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Techniques of. Water-Resources Investigations

of the United States Geological Survey

Chapter B7

ANALYTICAL SOLUTIONS FOR ONE-, TWO-,

AND THREE-DIMENSIONAL SOLUTE

TRANSPORT IN GROUND-WATER SYSTEMS

WITH UNIFORM FLOW

By Eliezer J. Wexler

• Book 3

APPLICATIONS OF HYDRAULICS

32169

1111111111111111111111111111111 I11I

U.S. DEPARTMENT OF THE INTERIOR

MANUEL LUJAN, Jr., Secretary

U.S. GEOLOGICAL SURVEY

Dallas 1. Peck, Director

UNITED STATES GOVERNMENT PRINTING OFFICE: 1992

For sale by Book and Open-File Report Sales, U.S. Geological Survey, Federal Center, Box 25425, Denver, CO 80225

PREFACE

The series of manuals on techniques describes procedures for planning and executing specialized work in water-resources investigations. The material is grouped under major subject headings called books and further subdivided into sections and chapters; section B of book 3 is on ground-water techniques.

The unit of publication, the chapter, is limited to a narrow field of subject matter. This format permits flexibility in revision and publication as the need arises. Chapter 3B7 deals with analytical solutions to the solute-transport equation for a variety of boundary condition types and solute-source configurations in one-, two-, and three-dimensional systems with uniform ground-water flow.

Provisional drafts of chapters are distributed to field offices of the U.S. Geological Survey for their use. These drafts are subject to revision because of experience in use or because of advancement of knowledge, techniques, or equipment. After the technique described in a chapter is sufficiently developed, the chapter is published and is for sale from the U.S. Geological Survey, Book and Open-File Report Sales, Federal Center, Box 25425, Denver, CO 80225.

Copies of the computer codes and sample data sets described in this report are available on diskette from Book and Open-File Report Sales as USGS Open File Report 92-78. They are on a 5.25” (360K) double-density diskette formatted for the IBM PC. The computer programs were originally written for a Prime minicomputer but all programs should run using IBM-PC Fortrans with minor modifications as described in the report. The plot routines were written with DISSPLA software calls and can be used on the PC only with the PC version of the DISSPLA library. Alternatively, data can be easily extracted from the program output and plotted using PC graphics presentation programs.

Reference to trade names, commercial products, manufacturers, or distribu- tors in this manual constitutes neither endorsement by the U.S. Geological Survey nor recommendation for use.

III

TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS OF THE U.S. GEOLOGICAL SURVEY

The U.S. Geological Survey publishes a series of manuals describing procedures for planning and conducting specialized work in water-resources investigations. The manuals published to date are list.ed below and may be ordered by mail from the U.S. Geological Survey, Books and Open-File Reports Section, Federal Center, Box 25425, Denver, Colorado 80225 (an authorized agent of the Superintendent of Documents, Government Printing Office).

Prepayment is required. Remittance should be sent by check or money order payable to U.S. Geological Survey. Prices are not included in the listing below as they are subject to change. Current prices can be obtained by writing to the USGS address shown above. Prices include cost of domestic surface transportation. For transmittal outside the U.S.A. (except to Canada and Mexico) a surcharge of 25 percent of the net bill should be included to cover surface transportation. When ordering any of these publications, please give the title, book number, chapter number, and “U.S. Geological Survey Techniques of Water-Resources Investigations.”

TWRI 1-Dl.

TWRI l-D2.

TWRI 2-Dl.

TWRI %D2. TWRI 2-El. TWRI 2-E2. TWRI 2-FL

TWRI 3-Al.

TWRI 3%A2. TWRI 3-A3. TWRI 3-A4. TWRI 3-A5. TWRI 3-A6. ‘IWRI 3-A7. TWRI 3-A8. TWRI 3-A9.’ TWRI 3-AlO. TWRI 3-All. TWRI 3-A12. TWRI 3-A13. TWRI 3-1114. TWRI 3-A15. TWRI 3-A16. TWRI 3-Al?. TWRI 3-AH.

TWRI 3-A19. TWRI 3-Bl. TWRI 3-B2.’

Water temperature-influential factors, field measurement, and data presentation, by H.H. Stevens, Jr., J.F. Ficke, and G.F. Smoot. 1975. 65 pages.

Guidelines for collection and field analysis of ground-water samples for selected unstable constituents, by W. W. Wood. 1976. 24 pages.

Application of surface geophysics to ground-water investigations, by A.A.R. Zohdy, G.P. Eaton, and D.R. Mabey. 1974. 116 pages.

Application of seismic-refraction techniques to hydrologic studies, by F.P. Haeni. 1988. 86 pages. Application of borehole geophysics to water-resources investigations, by W.S. Keys and L.M. MacCary. 1971. 126 pages. Borehole geophysics applied to ground-water investigations, by W. Scott Keys. 1990. 150 pages. Application of drilling, coring, and sampling techniques to test holes and wells, by Eugene Shuter and Warren E. Teasdale.

1989. 97 pages. General field and office procedures for indirect discharge measurements, by M.A. Benson and Tate Dalrymple. 1967.

30 pages. Measurement of peak discharge by the slope-area method, by Tate Dahymple and M.A. Benson. 1967. 12 pages. Measurement of peak discharge at culverts by indirect methods, by G.L. Bodhaine. 1968. 60 pages. Measurement of peak discharge at width contractions by indirect methods, by H.F. Matthai. 1967. 44 pages. Measurement of peak discharge at dams by indirect methods, by Harry Hulsing. 1967. 29 pages. General procedure for gaging streams, by R.W. Carter and Jacob Davidian. 1968. 13 pages. Stage measurements at gaging stations, by T.J. Buchanan and W.P. Somers. 1968. 28 pages. Discharge measurements at gaging stations, by T.J. Buchanan and W.P. Somers. 1969. 65 pages. Measurement of time of travel in streams by dye tracing, by F.A. Kilpatrick and J.F. Wilson, Jr. 1989. 27 pages. Discharge ratings at gaging stations, by E.J. Kennedy. 1984. 59 pages. Measurement of discharge by moving-boat method, by G.F. Smoot and C.E. Novak. 1969. 22 pages. Fluorometric procedures for dye tracing, Revised, by J.F. Wilson, Jr., E.D. Cobb, and F.A. Kilpatrick. 1986. 41 pages. Computation of continuous records of streamflow, by E.J. Kennedy. 1983. 53 pages. Use of flumes in measuring discharge, by F.A. Kilpatrick, and V.R. Schneider. 1983. 46 pages. Computation of water-surface profiles in open channels, by Jacob Davidian. 1984. 48 pages. Measurement of discharge using tracers, by F.A. Kilpatrick and E.D. Cobb. 1985. 52 pages. Acoustic velocity meter systems, by Antonius Laenen. 1985. 38 pages. Determination of stream reaeration coefficients by use of tracers, by F.A. Kilpatrick, R.E. Rathbun, N. Yotsukura, G.W.

Parker, and L.L. DeLong. 1989. 52 pages. Levels at streamflow gaging stations, by E.J. Kennedy. 1990. 31 pages. Aquifer-test design, observation, and data analysis, by R.W. Stallman. 1971. 26 pages. Introduction to ground-water hydraulics, a programmed text for self-instruction, by G.D. Bennett. 1976. 172 pages.

‘This manual is a revision of “Measurement of Time of Travel and Dispersion in Streams by Dye Tracing,” by E.F. Hubbard, F.A. Kilpatrick, L.A. Martens, and J.F. Wilson, Jr., Book 3, Chapter A9, published in 1982.

“Spanish translation also available. 0

IV

TWRI 3-B3. TWRI 3-B4. TWRI 3-B5.

TWRI 3-B6.

TWRI 3-B7.

TWRI 3-Cl. TWRI 3-C2. TWRI 3-C3. TWRI 4-Al. TWRI 4-A2. TWRI 4-Bl. TWRI 4-B2. TWRI 4-B3. TWRI 4-Dl. TWRI 5-Al.

TWRI 5-A2. TWRI 5-A3.’

TWRI 5-A4.’

TWRI 5-A5.

TWRI 5-A6.

TWRI 5X1. TWRI 6-Al.

TWRI 6-A2.

TWRI 7-Cl.

TWRI 7-C2.

TWRI 7-C3.

TWRI 8-Al. TWRI 8-A.2 TWRI 8-B2.

V

Type curves for selected problems of flow to wells in confined aquifers, by J.E. Reed. 1980. 106 pages. Regression modeling of ground-water flow, by Richard L. Cooley and Richard L. Naff. 1990. 232 pages. Definition of boundary and initial conditions in the analysis of saturated ground-water flow systems-An introduction, by

0. Lehn Franke, Thomas E. Reilly, and Gordon D. Bennett. 1987. 15 pages. The principle of superposition and its application in ground-water hydraulics, by Thomas E. Reilly, 0. Lehn Franke, and

Gordon D. Bennett. 1987. 28 pages. Analytical solutions for one-, two-, and three-dimensional solute transport in ground-water systems with uniform flow, by

Eliezer J. Wexler. 1991. 193 pages. Fluvial sediment concepts, by H.P. Guy. 1970. 55 pages. Field methods of measurement of fluvial sediment, by H.P. Guy and V.W. Norman. 1970. 59 pages. Computation of fluvial-sediment discharge, by George Porterfield. 1972. 66 pages. Some statistical tools in hydrology, by H.C. Riggs. 1968. 39 pages. Frequency curves, by H.C. Riggs, 1968. 15 pages. Low-flow investigations, by H.C. Riggs. 1972. 18 pages. Storage analyses for water supply, by H.C. Riggs and C.H. Hardison. 1973. 20 pages. Regional analyses of streamflow characteristics, by H.C. Riggs. 1973. 15 pages. Computation of rate and volume of stream depletion by wells, by C.T. Jenkins. 1970. 17 pages. Methods for determination of inorganic substances in water and fluvial sediments, by Marvin J. Fishman and Linda C.

Friedman, editors. 1989. 545 pages. Determination of minor elements in water by emission spectroscopy, by P.R. Barnett and E.C. Mallory, Jr. 1971.31 pages. Methods for the determination of organic substances in water and fluvial sediments, edited by R.L. Wershaw, M.J.

Fishman, R.R. Grabbe, and L.E. Lowe. 1987. 80 pages. Methods for collection and analysis of aquatic biological and microbiological samples, by L.J. Britton and P.E. Greeson,

editors. 1989. 363 pages. Methods for determination of radioactive substances in water and fluvial sediments, by L.L. Thatcher, V.J. Janzer, and

K.W. Edwards. 1977. 95 pages. Quality assurance practices for the chemical and biological analyses of water and fluvial sediments, by L.C. Friedman and

D.E. Erdmann. 1982. 181 pages. Laboratory theory and methods for sediment analysis, by H.P. Guy. 1969. 58 pages. A modular three-dimensional finite-difference ground-water flow model, by Michael G. McDonald and Arlen W. Harbaugh.

1988. 586 pages. Documentation of a computer program to simulate aquifer-system compaction using the modular finite-difference

ground-water flow model, by S.A. Leake and D.E. Prudic. 1991. 68 pages. Finite difference model for aquifer simulation in two dimensions with results of numerical experiments, by P.C. Trescott,

G.F. Pinder, and S.P. Larson. 1976. 116 pages. Computer model of two-dimensional solute transport and dispersion in ground water, by L.F. Konikow and J.D.

Bredehoeft. 1978. 90 pages. A model for simulation of flow in singular and interconnected channels, by R.W. Schaffranek, R.A. Baltzer, and D.E.

Goldberg. 1981. 110 pages. Methods of measuring water levels in deep wells, by M.S. Garber and F.C. Koopman. 1968. 23 pages. Installation and service manual for U.S. Geological Survey monometers, by J.D. Craig. 1983. 57 pages. Calibration and maintenance of vertical-axis type current meters, by G.F. Smoot and C.E. Novak. 1968. 15 pages.

‘This manual is a revision of TWRI 5-A3, “Methods of Analysis of Organic Substances in Water,” by Donald F. Goerlitz and Eugene Brown, published in 1972.

‘This manual supersedes TWRI 5-A4, “Methods for collection and analysis of aquatic biological and microbiological samples,” edited by P.E. Greeson and others, published in 1977.

CONTENTS

Page

Abstract ........................................................................ 1 Introduction.. ................................................................. 1

Purpose and scope .................................................... 2 Previous studies ....................................................... 2 Acknowledgments ..................................................... 2

Theoretical background.. .................................................. 2 Advection ................................................................ 2 Molecular diffusion .................................................... 3 Mechanical dispersion.. .............................................. 3 Hydrodynamic dispersion.. ......................................... 4 Advective-dispersive solute-transport equation.. ............ 4 Chemical transformation ............................................ 5

Linear equilibrium adsorption ............................... 5 Ion exchange ..................................................... 6 First-order chemical reactions ............................... 6

Initial conditions.. ..................................................... 7 Boundary Conditions ................................................. 7

Inflow boundary ................................................. 7 Outflow boundary ............................................... 8 Lateral boundaries.. ............................................ 8

Superposition ........................................................... One-dimensional solute transport .......................................

Finite system with first-type source boundary condition ........................................................

Governing equation.. ........................................... Analytical solution .............................................. Description of program FINITE ...........................

Main program .............................................. Subroutines ROOT1 and CNRMLl ..................

Sample problems la and lb .................................. Finite system with third-type source boundary

condition ........................................................ Governing equation ............................................. Analytical solution .............................................. Description of program FINITE ...........................

Subroutines ROOT3 and CNRML3.. ................ Sample problem 2 ...............................................

Semi-infinite system with first-type source boundary condition ........................................................

Governing equation ............................................. Analytical solution .............................................. Description of program SEMINF ..........................

Main program .............................................. Subroutine CNRMLl ....................................

Sample problems 3a and 3b .................................. Semi-infinite system with third-type source boundary

condition ........................................................ Governing equation ............................................. Analytical solution .............................................. Description of program SEMINF ..........................

Subroutine CNRML3 ....................................

8 11

12 12 12 13 13 13 13

17 17 17 18 18 18

18 18 18 20 20 20 20

21 21 21 24 24

Page

One-dimensional solute transport-Continued Semi-infinite system with third-type source boundary

condition-Continued Sample problem 4 ............................................... 24

Two-dimensional solute transport ...................................... 24 Aquifer of infinite area1 extent with continuous point

source ............................................................ 26 Governing equation ............................................. 26 Analytical solution .............................................. 27 Description of program POINT2 ........................... 27

Main program .............................................. 27 Subroutine CNRML2 .................................... 31

Sample problem 5 ............................................... 31 Aquifer of finite width with finite-width solute source ..... 31

Governing equation ............................................. 31 Analytical solution .............................................. 33 Description of program STRIPF ........................... 33

Main program .............................................. 33 Subroutine CNRMLF .................................... 33

Sample problem 6 ............................................... 33 Aquifer of infinite width with finite-width solute

source.. .......................................................... 36 Governing equation ............................................. 36 Analytical solution .............................................. 36 Description of program STRIP1 ............................ 36

Main program .............................................. 36 Subroutine CNRMLI .................................... 37

Sample problem 7 ............................................... 38 Aquifer of infinite width with solute source having

gaussian concentration distribution ..................... 38 Governing equation ............................................. 38 Analytical solution .............................................. 38 Description of program GAUSS ............................ 40

Main program .............................................. 40 Subroutine CNRMLG .................................... 40

Sample problems 8a and 8b .................................. 40 Three-dimensional solute transport .................................... 42

Aquifer of infinite extent with continuous point source .... 42 Governing equation ............................................. 42 Analytical solution .............................................. 43 Description of program POINT3 ........................... 47

Main program .............................................. 47 Subroutine CNRML3 .................................... 47

Sample problem 9 ............................................... 47 Aquifer of finite width and height with finite-width and

finite-height solute source ................................. 49 Governing equation ............................................. 49 Analytical solution .............................................. 51 Description of program PATCHF .......................... 51

Main program .............................................. 51 Subroutine CNRMLF .................................... 51

VII

VIII CONTENTS

Page

Three-dimensional solute transport-Continued Aquifer of finite width and height with finite-width and

finite-height solute source-continued Sample problem 10.. ............................................ 53

Aquifer of infinite width and height with finite-width and finite-height solute source ........................... 53

Governing equation ............................................. 53 Analytical solution .............................................. 53 Description of program PATCHI.. ......................... 55

Main program .............................................. 55 Subroutine CNRMLI .................................... 55

Sample problem ll.............................................. 57 Description of subroutines.. .............................................. 57

Mathematical subroutines.. ......................................... 57 Subroutines EXERFC and GLQPTS ..................... 57

Page Description of subroutines-Continued

Input/Output s&routines.. ......................................... 59 Subroutines OFILE and TITLE ........................... 59

Graphics subroutines ................................................. 59 Subroutines PLOTlD, PLOT2D, PLOT3D, and

CNTOUR .................................................... 59 Running the programs.. ............................................. 60

Array dimensions ............. .:. ................................ 60 Compiling and loading ......................................... 60

Summary ...................................................................... 61 References cited.. ............................................................ 61 Attachment 1. -Derivation of selected analytical solutions ..... 63 Attachment e--Program source-code listings.. ..................... 81 Attachment 3-Subroutine listing and data file GDQ.PTS ...... 135 Attachment I-Program output for selected sample

problems .............................................................. 163

1. 2. 3.

4-6.

7-9.

10.

11.

12-14.

15.

16, 17.

FIGURES

Graph showing typical shape of equilibrium adsorption isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Diagrams showing examples of situations in which the principle of superposition can be applied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Diagrams showing two examples of contaminant movement in field settings that can be simulated as one-dimensional

solute-transport systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... 11 Representations showing sample input data set and computer plot of concentration profiles generated by the program

FINITE for a: 4. Conservative solute in a finite-length system with first-type source boundary condition after 2.5, 5, 10, 15, and

20 hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................... 15 5. Solute subject to linear adsorption in a finite-length system with first-type source boundary condition after 20, 50,

100, and 150 hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... 16 6. Conservative solute in a finite-length system with third-type source boundary condition after 2.5, 5, 10, 15, and

20 hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *.....*. ..* . . . . . . . . . . . ,.......... . . . . . . . . . . . . . . . . . . . . . . 19 Representations showing sample input data set and computer plot of concentration profiles generated by the program

SEMINF for a: 7. Conservative solute in a semi-infinite system with first-type source boundary condition after 2.5, 5, 10, 15, and

20 hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................... 22 8. Solute subject to first-order decay and linear equilibrium adsorption in a semi-infinite system with first-type

source boundary condition after 20, 50, 100, and 150 hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 9. Conservative solute in a semi-infinite system with third-type source boundary condition after :!.5, 5, 10, 15, and

20 hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................... 25 Diagrams depicting plan view and vertical section of idealized two-dimensional solute transport in an aquifer of semi-infinite

length and finite width, and plan view of idealized two-dimensional solute transport in an aquifer of semi-infinite length and infinite width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................... 28

Diagram depicting plan view of a semi-infinite aquifer of infinite width showing location of waste-disposal pond and monitoring wells, and graph of observed solute concentration values and gaussian curve used to approximate concentration distribution at x=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Representations showing sample input data set and computer plot of normalized concentration contours generated by the program: 12. POINT2 for a conservative solute injected continuously into an aquifer of infinite area1 extent after 25 and

100 days . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................... 32 13. STRIPF for a conservative solute in an aquifer of finite width with finite-width solute source after 1,500 and

3,000 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . 35 14. STRIP1 for a conservative solute in an aquifer of infinite width with finite-width solute source after 1,826 days . . . . . . . 39

Histogram of normalized concentrations in relation to distance for the waste-disposal site in sample problem 8a and fitted gaussian distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................ 43

Representations showing sample input data set and computer plot of normalized concentration contours generated by the program GAUSS for a: 16. Conservative solute in an aquifer of infinite width having a gaussian concentration distribution (a= 150 feet) at the l .- . . 1 . intlow boundary at 300 days . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * * . . * . . . . . . . . . . . . . . . . . . . . . . . . 44

CONTENTS IX

Page

17. Conservative solute in an aquifer of infinite width having a gaussian concentration distribution (0=65 feet) at the inflow boundary at 300 days . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , ., . . . . . . , ,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

18. Diagrams showing plan view and vertical section of idealized three-dimensional transport in an aquifer of semi-infinite length and finite width and height . . . . .., . . . . . . , ,.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .., . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . 46

19-21. Representations showing sample input data set and computer plot of normalized concentration contours generated by the program: 19. POINT3 for a natural gradient tracer test in an aquifer of infinite extent after 400 days in the z = lo-foot plane., , . . . 50 20. PATCHF for a conservative solute in an aquifer of finite height and width after 3,000 days at heights of ‘75 and

50 feet above base of the aquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..I........................................ 54 21. PATCH1 for a solute subject to first-order chemical transformation in an aquifer of infinite height and width with

solute source of finite height and width after 3,652.5 days at heights of 1,650, 1,700, and 1,750 feet above base of the aquifer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

TABLES

Page l-6. Input data format for the program:

1. FINITE .................................................................................................................................................. 14 2. SEMINF ................................................................................................................................................. 21 3. POINT2 .................................................................................................................................................. 30 4. STRIPF .................................................................................................................................................. 34 5. STRIP1 ................................................................................................................................................... 37 6. GAUSS ................................................................................................................................................... 41

7. Measured solute concentrations in monitoring wells downgradient from the waste-disposal site in sample problem 8b ............ 42 3-10. Input data format for the program:

8. POINT3 .................................................................................................................................................. 48 9. PATCHF ................................................................................................................................................. 52

10. PATCHI.. ................................................................................................................................................ 56

METRIC CONVERSION FACTORS

For readers who wish to convert measurements from the inch-pound system of units to the metric system of units, the conversion factors are listed below: Multiply inch-pound units BY

inch (in) 25.4 To obtain metric units millimeter (mm)

inch per hour (k&j foot (ft)

foot per day (ftld) gallon (gal)

square inch per hour (in /h)

foot squared per day (ft’/d)

cubic foot (ft3) 0.02832 cubic foot per day (ft3/d) 0.02832

pound per cubic foot (lb/ft3) 0.01602

25.4 0.3048 0.3048 0.003785 6.4516

0.09290

millimeter per hour (mm/h) meter (m) meter per day (m/d) cubic meter (m3) square centimeter per hour

(cm%) meter

(m2/d) squared per day

cubic meter (m3) cubic meter per day (m3/d) gram per cubic centimeter

(g/cm3)

CONTENTS

DEFINITION OF SYMBOLS

Symbol

c

CO CEC

E P g

D* D,,D,,D, erf erfc ew h d

k =K K, KO n P P R S t T T II2 V

v,,vy,v,

%

V’ Z

a1 % Y 0 h P

2

: a

Dimension Dejkitions

M/L3 Volumetric concentration M/L3 Specified concentration value along a boundary l/M Cation exchange capacity L2PI Coefficient of hydrodynamic dispersion for one-dimensional solute transport L21T Hydrodynamic dispersion tensor in an isotropic system L2/T Molecular diffusion tensor L21T Mechanical dispersion tensor L2/T Dispersion coefficient divided by the retardation factor, R L2/T Magnitudes of the hydrodynamic dispersion tensor in a system having uniform flow - Error function - - L M/L2T L3/M L3/M LIT -

Complimentary error function Exponential Head

- - MILT2 - - - T - T L/T LIT L/T LiT LIT L L L M./L?! - l/T M/L3 M/L3 - - - -

Solute flux due to dispersion or diffusion Slope of the linear equilibrium adsorption isotherm Distribution coefficient Hydraulic conductivity tensor Selectivity coefficient for ion exchange reactions Modified Bessel function of second kind and zero order Effective porosity Fluid pressure Column Peclet number Retardation factor for adsorbed solute Mass concentration of adsorbed solute Time Number of displaced pore volumes Half-life of radioactive solute Magnitude of the average interstitial fluid velocity Magnitudes of the average interstitial velocity components Magnitude of uniform velocity Average interstitial fluid velocity Average interstitial fluid velocity divided by the retardation factor, R Elevation head Longitudinal dispersivity Transverse dispersivity Specific weight of water Soil moisture content (equal to porosity for saturated soils) First-order chemical transformation rate Density of water Bulk density of aquifer material Summation Infinity Gradient operator Partial derivative

ANALYTICAL SOLUTIONS FOR ONE-, TWO-, AND THREE-DIMENSIONAL SOLUTE TRANSPORT IN GROUND-WATER

SYSTEMS WITH UNIFORM FLOW

By Eliezer J. Wexler

Introduction

Contamination of ground water by inorganic and organic chemicals has become an increasing concern in recent years. These chemicals enter the ground-water system by a wide variety of mechanisms, including accidental spills, land disposal of domestic and indus- trial waste, and application of agricultural fertilizers and pesticides. Once introduced into an aquifer, these solutes will be transported by flowing ground water and may degrade water quality at nearby wells and streams.

Abstract

Analytical solutions to the advective-dispersive solute-transport equation are useful in predicting the fate of solutes in ground water. Analytical solutions compiled from available literature or derived by the author are presented for a variety of boundary condition types and solute-source configurations in one-, two-, and three- dimensional systems having uniform ground-water flow. A set of user-oriented computer programs was created to evaluate these solutions and to display the results in tabular and computer- graphics format. These programs incorporate many features that enhance their accuracy, ease of use, and versatility. Documentation for the programs describes their operation and required input data, and presents the results of sample problems. Derivations of selected solutions, source codes for the computer programs, and samples of program input and output also are included.

To improve management and protection of ground- water resources, it is important to first understand the physical, chemical, and biological processes that control the transport of solutes in ground water. Predictions of the fate of ground-water contaminants can then be made to assess the effect of these chemicals on local water resources and to evaluate the effectiveness of remedial actions.

Two physical processes that govern the movement of ground-water solutes are (1) advection, which describes the transport of solutes by the bulk motion

of flowing ground water (Freeze and Cherry, 1979), and (2) hydrodynamic dispersion, which describes the spread of solutes along and transverse to the direction of flow resulting from both mechanical mixing and molecular diffusion (Bear, 1979, p. 230). Chemical reactions, including those mediated by microorgan- isms or caused by interaction with aquifer material or other solutes, may also affect the concentration of the solute.

These prqcesses have been described quantitatively by a partial differential equation referred to as the “advective-dispersive solute-transport equation.” Solution of the equation yields the solute concentration as a function of time and distance from the contaminant source. To apply the equation to a particular ground-water contamination problem, data must be provided on the ground-water velocity, coefficients of hydrodynamic dispersion, rates of chemical reactions, initial concentrations of solutes in the aquifer, configuration of the solute source, and boundary conditions along the physical boundaries of the ground-water flow system.

In ground-water systems having irregular geometry and nonuniform aquifer properties, numerical techniques are used to determine approximate solutions to the solute-transport equation. In aquifers having simple flow systems and relatively uniform hydrologic properties, analytical solutions, which represent exact mathematical solutions to the solute- transport equation, have been used to predict contaminant migration. These solutions are also used extensively in analysis of data from soil-column experiments and field tracer tests to determine aquifer properties, and have been used to verify the sound- ness of numerical models. In complex hydrogeologic systems, analytical solutions can still be useful to the hydrologist because they can provide estimates of

1

2 TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS

rates of solute spread and, thus, guide data collection and water-quality-monitoring efforts.

Although deriving an analytical solution for the solute-transport equation requires knowledge of higher mathematics, analytical solutions have already been derived and published for many combinations of solute-source configurations and boundary-condition types. After the solutions have been derived, they can be evaluated easily using electronic calculators or digital computers.

Purpose and scope This report briefly describes the theoretical back-

ground of solute transport in a porous medium and then presents analytical solutions to the advective- dispersive solute-transport equation for a variety of aquifer and solute-source configurations and boundary conditions in systems having uniform (unidirectional) ground-water flow. Solutions for one-dimensional solute transport were compiled from various journals and reports, many of which are not readily available. Many of the solutions for two- and three-dimensional solute transport were modified from those presented in a report by Cleary and Ungs (19’78), whereas others were derived by the author using integral transform techniques. (Detailed derivations of these solutions are provided in attachment 1.) All solutions are given in a simplified format, together with information on important assumptions in their derivation and limita- tions to their use.

Simple computer programs, written in FORTRAN- 77, have been provided for evaluation of the analytical solutions presented. The programs were designed for ease of use and for enhanced accuracy. Documentation for these programs includes descriptions of program operation and the input data required. Source codes and samples of program output are provided at the end of the report. Subroutines that allow for graphical display of the program output, created using DISS- PLA software, are also described. Computer- generated plots are presented within the report.

Previous studies Analytical solutions for the one-dimensional form of

the solute-transport equation have appeared in reports and journals concerning physical chemistry, soil science, and water resources. These solutions, generally determined through Laplace transform techniques, have been applied to studies of solute movement in laboratory columns, unsaturated soils, and natural-gradient tracer tests. Solutions to the one-dimensional solute-transport equation for most

combinations of boundary and initial conditions are given in van Genuchten and Alves (1982); some of the more useful solutions appear in this report. Other sources that list several analytical solutions include Gershon and Nir (1969), Bear (19’72), and Bear (1979).

Fewer analytical solutions have been published for the two- and three-dimensional forms of the solute- transport equation. Cleary and IJngs (1978) give several solutions derived using integral transform techniques, and Yeh (1981) presents a computer program that evaluates Green’s function to model one-, two-, and three-dimensional transport.

Acknowledgments This report was prepared at the request of Thomas

E. Reilly of the U.S. Geological Survey, Office of Ground Water, in Reston, Va., who recognized the need for a compilation of analytical solutions to the transport equation. Thanks are extended to Robert Cleary, formerly of Princeton University, and Edward Sudicky of the University of Waterloo, Ontario, Canada, for introducing the author to the application of analytical solutions to the analysis of water-quality problems. Thanks are also extended to Mary Hill, Daniel J. Goode and Stephen P. Garabedian of the U.S. Geological Survey for their comments on an earlier version of this report.

Theoretical Background

Most models that simulate migration of dissolved contaminants in ground water solve the advective- dispersive solute-transport equation. This partial differential equation is derived from the conservation- of-mass principle (continuity equation), whereby the net rate of change of solute mass within a volume of porous media is equal to the difference between the flux of solute into and out of the volume, adjusted for the loss or gain of solute mass due to chemical reactions (Freeze and Cherry, 1979). The flux of solute into the volume is controlled by two physical processes-advection and hydrodynamic dispersion. Hydro- dynamic dispersion, in turn, represents the combined effects of two other physical processes-molecular diffusion and mechanical dispersion.

Advection Advective transport describes the bulk movement

of solute particles along the mean direction of fluid flow at a rate equal to the average interstitial fluid velocity. In a saturated medium, this velocity can be calculated from Darcy’s law, such that

ANALYTICAL SOLUTIONS FOR SOLUTE TRANSPORT IN GROUND-WATER SYSTEMS WITH UNIFORM FLOW 3

where (1)

2 =average interstitial fluid velocity [L/T], K = hydraulic conductivity tensor for medium

WJJ, n =effective porosity [dimensionless], and

& =gradient in head [dimensionless] (equal to dhldx, the change in head per unit distance along the x-axis for uniform flow along the x-axis).

Head, h in equation 1, is equal to the sum of the elevation head, z, with respect to a datum level, and the pressure head, p/y, where p is the fluid pressure (gage pressure) and y is the specific weight of water (Bear, 1979, p. 62). Water flows from areas of higher head toward areas of lower head. Effective porosity, n, differs from total porosity (volume of pore space per unit volume of aquifer material) in that it does not include pores that are too small to transmit water or “dead-end” pores, those that are not interconnected with other pores.

In unsaturated porous media, the average interstitial fluid velocity can be approximated (Bear, 1979, p. 209) as

where v’ =average interstitial fluid velocity [L/T],

z(0) =unsaturated hydraulic conductivity tensor for medium, which is a function of moisture content [LPI’], and

8 =moisture content of soil [dimensionless]. This form of the equation assumes that the movement of air in the soil is negligible and that the density of water is constant.

Molecular diffusion

In addition to advective transport, solutes spread within the fluid in the porous medium by molecular diffusion. Diffusion results from the random collision of solute molecules and produces a flux of solute particles from areas of higher to lower+solute concentration (Bear, 1979). The solute flux, J, can be given by Fick’s first law as

(W

whzre Dd = second-rank diagonal tensor of molecular diffu-

sion lL’/T],

Scheidegger (1961) stated that the coefficients of mechanical dispersion can be related to the average interstitial fluid velocity by means of the geometric dispersivity of the medium. For a saturated porous medium, geometric dispersivity depends on hydraulic conductivity, length of a characteristic flow path, and tortuosity (Bear, 1972, p. 614). In a medium that is isotropic with respect to dispersion, geometric dispersivity can be expressed in terms of just two coefficients-longitudinal dispersivity, al, and transverse dispersivity, tit (Bear, 1979, p. 234).

+C =concentration gradient [M/L4], and The elements of the mechanical dispersion tensor C =concentration of solute (mass of solute per unit can be expressed in terms of longitudinal and trans-

volume of fluid) [M/L3]. verse dispersivities, the magnitude of the velocity

Bear and Bachmat (1967) state that the coefficients of molecular diffusion in an isotropic medium are dependent on the diffusion coefficient of the particular solute in water and the tortuosity of the medium. Rates of molecular diffusion are independent of ground-water velocity, and diffusion occurs even in the absence of fluid movement.

Mechanical dispersion

The average interstitial fluid velocity represents a mathematical approximation. True velocities at points in the aquifer will differ from this average value, in both magnitude and direction. Local variations in ground-water velocity may not greatly affect the bulk movement of ground water, but they do control the fate of solute particles.

Mechanical dispersion describes the mixing and spreading of solutes along and transverse to the direction of flow in response to local variations in interstitial fluid velocities. On a microscopic scale (the scale of individual pores), mechanical dispersion results from (1) the distribution of velocities within an individual pore due to friction effects along the surface of soil grains, (2) differences in size of pores, (3) differences in path length for individual solute particles, and (4) the effect of converging and diverging flow paths (Freeze and Cherry, 1979, p. 75). On a larger (macroscopic) scale, mechanical dispersion results from local variations in hydraulic conductivity, and thus fluid velocity, owing to the heterogeneity of aquifer material (Bear, 1979, p. 229).

Laboratory tests on soil columns have shown that the flux of solutes due to mechanical dispersion can also be described using Fick’s first law, as

f= -tGm.+c, CW

where D, is the second-rank symmetric tensor of mechanical dispersion [L’/T].


vector, v, and the magnitudes of its components, vX, vY, and v, (Bear, 1979, p. 235) as

D mn =[~*v,2+olt(Vy2+V,2)1/V D mYY =[c(p,2+a,(v,2+v,2)1/v D

mxY =Dmfl =(cYpt)vxvy/v

D my2 =DmZy =(ctp.t)VyV~v

D =D DlI= [‘yIvZ

m~=(“,-“thxVz/V + (Y&v,~ +vy2)l/v. (4)

If a coordinate system is chosen, such that the direction of the average ground-water velocity is aligned with the x-direction (v=v, and vy=vZ=O), the off-diagonal terms in the dispersion tensor (eq. 4) will equal zero, and the mechanical dispersion tensor can be simplified to

Dmx=Dmxx=~l~ Dmy=Dm =r+v DmZ=DmZy=~t~. (5)

Hydrodynamic dispersion As stated earlier, hydrodynamic dispersion is the

flux of solute due to the combined effect of molec$ar diffusion and mechanical dispersion. Solute flux, J, is given by Fick’s first law as

f= -eE&, (6) zzz

where D is the hydrodynamic dispersion tensor. In a flow system having uniform flow aligned with

the x-axis, the coefficients of the hydrodynamic dispersion tensor, D,, D,, and D,, are given by

Dx=DmX+Dd=~I~+Dd Dy=DmY+Dd=atv+Dd D,=DmZ+Dd=atv+Dd. (7)

The effects of mechanical dispersion generally are much greater than those of molecular diffusion, and, except at low ground-water velocities, the contribu- tion of molecular diffusion often is negligible.

In laboratory experiments using homogeneous materials, values for longitudinal dispersivity, al, are typically between 0.004 and 0.4 inch (in), whereas in field studies, longitudinal dispersivities of as much as 328 feet (ft) have been determined (Freeze and Cherry, 1979). The larger field values can be attrib- uted to increased mixing due to local variations in hydraulic conductivity (macrodispersion). A discus- sion of the apparent scale dependency of hydrodynamic dispersion is given in Anderson (1984). Trans-

verse dispersivity is generally less than longitudinal dispersivity, by a factor of 5 to 20 (Freeze and Cherry, 1979, p. 400).

Advective-dispersive solute-transport equation1

The advective-dispersive solute-transport equation describes the time rate of change of solute concentration for a single solute and can be written as

aec ~=-~*[e;C-eD=*~Cl+eQ, (8)

where Q, is used to represent a general source or sink term for production or loss of solute within the system.

Equation 8 (after Bear, 1979, p. 241) can be written in terms of volumetric rather than mass concentrations because the fluid density is assumed to be constant. This is usually valid for most ground-water flow systems in which solutes are present in relatively low concentrations.

The analytical solutions presented in this report are derived for idealized systems in which the ground- water velocity is assumed to be uniform, aligned with the x-axis, and of constant magnitude. The moisture content (equal to porosity for satnrated material) and the coefficients of hydrodynamic dispersion (see eq. 7) are also assumed to be constant. Given these assumptions, the three-dimensional form of the solute- transport equation for a uniform flow system can be expressed as

aceD a2C z-

x,+Dy$+D,$-V$+Q~, (9) 1,

where V represents the uniform velocity aligned with the x-axis.

In a thin aquifer in which the solute is uniformly mixed in the vertical (y-z) plane at the inflow boundary, the concentration gradient. in the z-direction, aC/az, equals zero. The two-dimensional solute- transport equation can be expressed as

Finally, if the solute concentration is uniform over the entire inflow boundary, such as in a soil column, the term aC/ay would also equal zero, yielding the one-dimensional solute-transport equation that can be expressed as

ac a% ac -g=Dz-Vz -t&s, (11)


where D represents the dispersion coefficient along the direction of flow.

Chemical transformation

In addition to physical mechanisms that govern the movement of solutes through the ground-water system, chemical transformations may alter the concentration of a contaminant species in solution. Possible chemical transformations include dissolution, precipi- tation, oxidation, reduction, biological degradation, radioactive decay, and adsorption and ion-exchange reactions between the solute and the solid matrix of the aquifer.

If the processes involved in chemical transformation can be described mathematically, they then can be incorporated in the source term, Q,, in the solute- transport equation for each chemical species. The analytical solutions described herein have been derived for systems in which the chemical transformation terms are given by first-order (linear) relations. The relations and their incorporation in the solute- transport equation are described below.

linear equilibrium adsorption

Many ionic inorganic solutes and nonpolar organic solutes can be removed from solution by adsorption onto the surface of soil particles. The solute may be attracted to soil surfaces by either electrical attrac- tion, Van der Wals forces, or chemical bonding (chemi- sorption). A general expression for the change in solute concentration due to partitioning of solute particles on the solid matrix (in the absence of dispersive or advective fluxes) can be stated as

ac as eat= - Pbx f (1%

where pb =bulk density of solid matrix measured as mass

per unit volume of aquifer material [M/L3], and S =mass of solute adsorbed on solid matrix per

unit mass of solid material [dimensionless]. The amount of solute remaining in solution depends

on the amount of solute in the adsorbed phase. The functional relation is usually determined experimentally through a series of batch tests in which solutions of known initial concentration are mixed with differing amounts of adsorbate. After equilibrium is achieved, the final solute concentration of each solution is measured, and the mass of solute adsorbed is calculated. An equilibrium adsorption curve can then be fitted to these data. Equilibrium concentrations are dependent on temperature, and the adsorption curve at a partic-

Observed mass concentration vsluell

lo.‘0 I I I I I

IO.8 10-G 10” 10” Id IO” 10’.

CONCENTRATION OF SOLUTE IN SOLUTION. IN MILLIGRAMS PER LITER

Figure 1 .-Typical shape of equilibrium adsorption isotherm.

ular temperature is termed an “equilibrium adsorption isotherm.” A typical equilibrium adsorption isotherm is shown in figure 1.

A linear approximation of the equilibrium adsorption isotherm is generally applicable in systems in which the solute concentration is low relative to the adsorptive capacity of the porous medium. The adsorption of various nonionic organic solutes at trace concentrations onto sediments and soils has also been shown to be linear (Cherry and others, 1984). Many nonlinear forms for the adsorption isotherm, some empirical and some that account for the physical mechanisms of adsorption, are suggested in the literature (see Helfferich, 1962). However, the transport equation that incorporates these other forms must be solved by numerical methods.

Because the amount adsorbed depends solely on the solute concentration, equation 12 can be expressed as

(13)

where aS/aC is determined from the functional relation between C and S. For a linear equilibrium adsorption isotherm, &/Z is equal to the slope of the adsorption isotherm and often is termed k or the “partitioning coefficient.” The source term can be incorporated in the general three-dimensional form of the solute-transport equation (eq. 9) to yield

(14)

where R is referred to as the “retardation factor,” given by


Dividing through by R yields

where V* and D:, D;, and D: are the scaled (or retarded) velocity and dispersion coefficients, respectively. Equation 16 shows that transport of solutes subject to linear adsorption can be simulated in the same manner as a nonadsorbed solute using these scaled coefficients. Because the apparent velocity of the adsorbed solute is reduced, the solute will arrive at a given point later than a nonadsorbed solute.

The use of equilibrium isotherms assumes that equilibrium exists at all times between the porous medium and the solute in solution. This assumption is generally valid when the adsorption process is fast in relation to the ground-water velocity (Cherry and others, 1934). If adsorption proceeds slowly, kinetics of the reaction must be considered. Nonequilibrium adsorption relations can be incorporated in the transport equation, but numerical methods are needed for solution of the resulting equation.

The process of adsorption is also assumed to be reversible. If hysteresis effects during desorption are significant, other forms of the adsorption isotherms must be considered, and numerical methods would be required.

Ion exchange

Ion exchange is an adsorption process in which a cation in solution replaces another cation that is elec- trically bound to collodial material in the soil. Under certain conditions, ion exchange can be modeled in a manner similar to linear adsorption (R.W. Cleary, Princeton University, written commun., 1977). The exchange reaction for monovalent ions can be expressed as

A++B+RsB++A+R, (17)

where A+ is used to represent a cation in solution, R is the exchange medium, and B+ represents the counter ion released from the exchanger. At equilibrium, a selectivity coefficient, &, can be defined, such that

EB+I[A+Rl K=[A+,[B+R] ’ (18)

where the bracketed terms represent the activities of each constituent.

Measured values of K, can be used in simulating transport by making the following assumptions: (1) if all exchange sites are assumed to be occupied initially,

then [B+Rl represents the total cation exchange capacity (CEC) of the medium, which can be determined experimentally and then treated as a constant; (2) the counter ion, Bf, is usually present in solution at much greater concentrations than the solute A+, and releases of additional amounts of the counter ion by exchange will not significantly alter its concentration; thus, [B+l can also be treated as a constant; and (3) the relation between the amount of solute on the exchange sites and the amount remaining in solution can be defined as

(19)

where the distribution coefficient, kd, is determined through laboratory batch tests.

Given these assumptions, the general expression for the change in solute concentration due to cation exchange can be expressed as

ac aC eat= -P&x,

where

cw

(21)

This term would replace k in equation 15. For monovalent-divalent cation exchange, where

A+++2B+R*2Bfi-A++R, (22)

the distribution coefficient can be given by

K,*CEC” kd= [B+]2 ’ (2%

First-order chemical reactions

Simple chemical reaction terms’ can be formulated to account for the kinetics of reactions under nonequilibrium conditions. A first-order chemical process, such as radioactive decay or biological degradation, involves the irreversible unimolecular conversion of solute A to solute B (A+B). The rate of the reaction can be given by

(24

where A is the rate coefficient [Y/T]. The rate coefficient can be expressed in terms of the half-life of the solute, T,,, (the time required for the concentration of the solute species to be reduced to half the initial concentration), as


h=ln(2)/Tl~,=0.693/‘I’1~~. (25)

Equation 9 can be written to incorporate the first- order reaction as

ac a% a2C at=DXz+D

yv +.,$_,~-A,. (26)

If the solute is subject to linear adsorption and to first-order chemical transformation in both the solute and adsorbed phases, equation 9 can be expressed as

ac a2C R,t=D,,x,+D y$+D,$-V$RhC (27)

where R is the retardation factor (eq. 15) or

aC .a2C .a2C at- -D,=+D yv

+D;$-V*$-XC, (28)

where V’ and D” represent the scaled velocity and dispersion coefficients. If the adsorbed phase is not subject to chemical transformation, A in equation 28 should be replaced by A*, where

X.=X/R. (29)

Some multiple-ion reactions can be approximated as a first-order reaction if all ions, except the species being considered, are present at high concentrations (R.W. Cleary, Princeton University, written commun., 1977). For example, if the reaction involves the conversion of solutes A and B to form solute C(A+B +C>, the rate of reaction would be given as

d[Al -= -A,,[AI[B]. dt (30)

If solute B is present at high concentration, its concentration will not change significantly due to conversion of A and [B] can be treated as a constant. Equation 26 or 28 can then be used with a modified A term, where A=AAB[B]. General bimolecular or multiple-ion reactions result in nonlinear chemical source terms. Reversible reactions and multistep reactions require simultaneous solution of the transport equation written for each species. Simulation of transport involving these chemical processes usually requires numerical methods.

Initial conditions To solve the solute-transport equation, a complete

set of boundary and initial conditions must be specified. Initial conditions are used to define the solute concentration in the aquifer at the time inflow of solute begins. For the analytical solutions presented

in this report, the initial conditions are specified such that all initial concentrations are zero. If the solute is conservative, a constant initial background solute concentration can be added to the calculated concentrations. Analytical solutions for one-dimensional transport of nonconservative solute transport with nonzero initial concentrations are given in van Genu- chten and Alves (1982).

Boundary conditions

Three types of boundary conditions are generally associated with the solute-transport equation. The first-type (or Dirichlet) boundary condition specifies the value of the concentration along a section of the flow-system boundary. The second-type (or Neu- mann) boundary condition specifies the gradient in solute concentration across a section of the boundary. The third-type (or Cauchy) boundary condition is applied where the flux of solute across the boundary is dependent on the difference between a specified concentration value on one side of the boundary and the solute concentration on the opposite side of the boundary. These three types of boundary conditions are used to describe conditions at the inflow and outflow ends of the flow system and also along the lateral boundaries of two- and three-dimensional systems.

Inflow boundary

The third-type boundary condition best describes solute concentrations at the inflow end in a uniform flow system (Bear, 1979, p. 268), where a well-mixed solute enters the system by advection across the boundary and is transported away from the boundary by advection and dispersion. The boundary conditions can be given as

VC-D$=VC,, x=0,

where C, is the known measured concentration in the influent water. The third-type boundary condition allows for solute concentration at the inflow boundary to be lower than C, initially and then to increase as more solute enters the system. Over time, the concentration gradient across the boundary, X%x, decreases as the concentration at the inflow boundary approaches C,.

Alternatively, a first-type boundary condition can be specified at the inflow end, such that

c=c,, x=0. (3%

Application of this simpler form of boundary condition presumes that the concentration gradient across the


boundary equals zero as soon as flow begins. How- ever, this may lead to overestimation of the mass of solute in the system at early times.

Equation 31 indicates that the difference between concentrations predicted for a system having a first- type source boundary condition and a system having a third-type boundary condition should decrease as the quantity DN decreases. Additional discussions of the effect and relative merits of the different types of inflow boundary conditions are presented in Gershon and Nir (1969), van Genuchten and Alves (1982), and Parker and van Genuchten (1984).

Outflow boundary

Often, the outflow boundary of the system being simulated is far enough away from the solute source that the boundary will not affect solute concentrations within the area of interest. Such a system can be treated as being “semi-infinite,” and either a first-type or second-type boundary condition can be specified as

C, aC- a,-% x=m.

When the system has a finite length, and solute concentrations near the outflow boundary are of interest, selection of an appropriate boundary condition becomes more difficult. In general, if the system discharges to a large, well-mixed reservoir and the additional solute will not significantly alter reservoir concentrations, then a third-type or first-type boundary condition (similar to the inflow boundary) can be used. If the reservoir is small or not well mixed, such as at the end of the soil column in figure 2A, concentrations in the reservoir would equal solute concentration at the discharge end of the system, and thus no concentration gradient would exist across the boundary. This can be specified by a second-type boundary condition as

ac x=0, x=L,

where L represents the length of the finite system. Van Genuchten and Alves (1982, p. 90-96) analyzed

the difference between predicted concentrations obtained using analytical solutions for a semi-infinite system and a finite system having a second-type boundary condition in terms of two dimensionless numbers: (1) the column Peclet number (P) and (2) the number of displaced pore volumes (T), which are defined by

p,vL D

and C=Co*A(x,y,z,t)+(C1-C,)rA(x,y,z[t-tJ), (39)

(35)

T,Vt L’

They found that the predicted concentration at points near the outflow boundary begins to differ significantly for T greater than 0.25 and that the differences increase as T approaches 1 (corresponding to movement of the solute front closer to the outflow boundary). The magnitude of the difference and distance inward from the outflow boundary at which the solutions diverge decreases as P values increase.

lateral boundaries

In two- and three-dimensional systems, impermeable or no-flow boundaries may be present at the base, top, or sides of the aquifer. Because there is no advective flux across the boundary, and molecular diffusion across the boundary is assumed to be negligible, the general third-type boundary condition sim- plifies to a second-type boundary condition, expressed as

aC ay=o, y=O and y=W (374

and

8C z=o, z=O and z=H (3%)

where W and H represent the width and height of the aquifer, respectively.

In many cases, lateral boundaries of the system may be far enough away from the area of interest that the system can be treated as being infinite along the y- and z-axes. Boundary conditions can then be specified as

c aC=(-j 7 ay 7

y::+cQ (384

and

c aC,() 'az 7

z- *co. (38b)

Superposition Because the solute-transport equation is a linear

partial-differential equation, the principle of superposition can be used to calculate concentrations in the system if solute concentrations at the inflow boundary vary over time. The general form of the solution can be expressed as


Velocity (vkO.5 inch per hour

Fence line

C&o c=c, c=c* V V V

-- -- -- --

--. -- -- -- --- -

-- P Sample port at

x=10 inches

----a - x=L=lOO inches

Direction of ground-water flow

V=lrO foot per day

Private well at x=500 feet ~~300 feet

I, I I I m 0 250 500 X

DISTANCE ALONG X-AXIS, IN FEET

Solute concentration Co=100 milligrams per liter Cl=300 milligrams per liter C,=O.O milligram per liter

50 300 900

AVERAGE CONCENTRATION ALONG BOUNDARY, IN MILLIGRAMS PER LITER

Figure 2.-Examples of situations in which the principle of superposition can be applied: A, soil column with time-varying input concentration (cases A and B in text), B, waste-disposal site with spatially varying input concentrations (case C in text), and C, plot of average concentration measured along waste-disposal site boundary.


where C, =initial solute concentration at boundary,

A(x,y,z,t) =general form of analytical solution where concentration is a function of space and time,

C1 =solute concentration at boundary after t=t,, and

tl =time at which solute concentration chang- es at boundary.

The principle of superposition should be familiar to most hydrologists who have used analytical solutions (such as the Theis equation) in analyzing aquifer tests. Several examples are provided to illustrate its application to solute-transport simulation.

Case A:

A solution is passed through a loo-in-long soil column (fig. 2A) for a period of 10 hours, with V=O.5 inch per hour (in/h), D=0.05 square inch per hour (in’/h), and C,= 100 milligrams per liter (mg/L). At the end of the lo-hour period, the concentration of the influent is increased to C,=300 mg/L. Of interest is the concentration at x=10 in at the end of a total elapsed time of 20 hours.

The analytical solution for transport of a conservative solute in a semi-infinite column (assuming that boundary effects at x=L are negligible) with a first- type inflow boundary condition was given by Ogata and Banks (1961) as

*(x,t)=$erfc[s]+exp[!$]*erfc[$]],

where erfc is the complementary error function. (The solution is described in more detail later.) For the values given, equation 39 becomes

C(l0 in, 20 hours)=100 mg/L*A(lO in, 20 hours)+(300 mg/L-100 mg/L>A(lO in, [20 hours-10 hours])

=lOO mg/L*(0.984)+200 mg/L l (0.088>

=116.0 mg/L.

Case B:

This case is similar to case A, except that at the end of 10 hours, solute-free water (C,=O.O mg/L) is passed through the soil column, thus creating a solute pulse of finite duration. The concentration of solute at x= 10 in and t=20 hours can be given from equation 39 as

C(10 in, 20 hours)=100 mg/L*A (10 in, 20 hours)+ (O-100 mg/L>A (10 in, [20-10 hours])

=lOO mg/Le(0.984)-100 mg/L l (0.088)

=89.6 mg/L. The principle of superposition can also be used to

simulate more complex solute configurations at the boundary of two- and three-dimensional systems, as shown in case C. Also, if solute sources are at two locations, the calculated concentration from the first source at a particular point of interest can simply be added to the calculated concentration from the second source at that point.

Case C:

A waste-disposal site, shown in plan view in figure 2B, has a solid-waste landfill and a smaller area for sludge disposal. Measured concentrations in fully screened wells along the eastern boundary downgradient from the landfill had chloride concentrations averaging 300 mg/L. Wells downgradient from both the sludge pond and the landfill had concentrations averaging 900 mg/L. Background chloride concentrations are 50 mg/L. Given V=l foot per day (ft/d), D, =20 feet squared per day (ft’/d>, and D,=4 ft2/d, calculate the concentration at a private well located at x=500 ft and y=300 ft at the end of 1 year.

The analytical solution for transport of a conservative solute in an infinitely wide aquifer having a finite-width or “strip” source along the inflow boundary is modified from Clear-y and Ungs (1978, p. 17):

-erfc

where YI and Ya are coordinates of the endpoints of the source on the y-axis and T is a dummy variable of integration. The solute source can be represented by two strip sources, the first extending from Y1=200 to Y2=8OO ft, with an effective concentration, CI, of 250 mg/L (difference between measured and background concentration) and the second extending from YI= 400 to Ya=6OO ft, with a concentration, Ca, of 600 mg/L (measured concentration minus first-source effective concentration and background concentration). The concentration at the private well can be calculated as

c(500 ft, 300 ft)=Cbae~groun~+C10A(500 ft, 300 ft, 200 ft, 800 ft, 365 days)+Cz*A(500 ft, 300 ft, 400 ft, 600 ft, 365 days)


A

3 Land surface c=c,

Unsaturated zone

B

Not to scale

Figure 3.-Two examples (A and B) of contaminant movement in field settings that can be simulated as one-dimensional solute-transport systems.

=50 mg/L+250 mg/L*(O.1612)+600 developed for study of dispersion phenomena in soil or mg/L*(O. 1354) adsorption columns. Some field situations can also be

=171.5 mg/L. idealized as one-dimensional transport systems; two

One-Dimensional Solute examples are shown in figure 3. Figure 3A represents steady vertical flow through the unsaturated zone

Transport beneath a septic tank drain field. Transport at the center of the field is simulated, and the horizontal

Many analytical solutions for the one-dimensional spread of solutes along the edges of the field is form of the solute-transport equation (eq. 11) were ignored. Figure 3B represents a case of steady hori-


zontal ground-water flow from river A, which has been contaminated, to river B.

One-dimensional systems can be finite, semi- infinite, or infinite in extent. In the finite or semi- infinite systems, water containing a known concentration of a contaminant species enters the system at the origin (at x=0). Water and solute exit at the opposite end of the system (at x=L), which could represent the water table, a stream, or the end of a soil column (fig. 3).

In the finite-length system, the outflow boundary is close enough that it will have an effect on the magnitude of concentrations within the area of interest. If the outflow boundary is far enough away as to have negligible effect on solute concentrations in the area of interest (equivalent to T<0.25, where T is the number of displaced pore volumes), solutions for a semi- infinite system can be used and are generally easier to evaluate.

An example of transport in an infinite system might be the injection of a solute into the center of a long soil column. In this case, the spread of solute, both upgra- dient and downgradient from the source, is of interest. Solutions for an infinite system can be found in van Genuchten and Alves (1982) and Bear (1972, 1979).

For the four analytical solutions presented in this section, either a first- or third-type boundary condition is specified at the inflow end of a finite or semi-infinite system. Specifically, the solutions are for a

Finite system with a first-type boundary condition at the inflow end, Finite system with a third-type boundary condition at the inflow end, Semi-infinite system with a first-type boundary condition at the inflow end, and Semi-infinite system with a third-type boundary condition at the inflow end.

Solutions for the finite systems assume a second-type boundary condition at the outflow end.

Two computer programs, FINITE and SEMINF, were developed to calculate concentrations in these four systems as a function of distance and elapsed time. These programs are also described in this section. The format used in presenting each of the solutions may seem repetitive, but it provides for easy reference.

Finite system with first-type source boundary condition

Governing equation

One-dimensional solute-transport equation:

(40) l Boundary conditions:

aLo dx- ’

XEL

Initial condition:

c=o, o<x< L at t=O (43)

Assumptions :

1. 2.

3. 4.

Fluid is of constant density and viscosity. Solute may be subject to first-order chemical transformation (for a conservative solute, X=0>. Flow is in x-direction only, and velocity is constant. The longitudinal dispersion coefficient (D), which is equivalent to D, (eq. 7), is constant.

Analytical solution

The following equation is modified from van Genu- chten and Alves (1982, p. 63-65):

C(x,t)=C,

-2 exp[g+-g]

where U=flL+4XD and pi are the roots of the equation

VL p cot p+m=o. (45)

Comments:

Values of the first six roots of equation 45, a*cot(a) +c=O, are tabulated in Carslaw and Jaeger (1959, p. 492) for various values of the constant c. Additional roots of equation 45 can be found through standard root-search techniques.

The maximum number of terms that should be computed in the infinite series summation depends on


how fast the series converges. Convergence is usually a problem at early times (or at T << 1) near the origin (x=0), especially when the column Peclet number (P in eq. 35) is relatively large. The program described below determines that the series has converged if the absolute value of the last term in the series is less than 1x10-12. A good initial estimate for the maximum number of terms is 100, but more should be used if the program indicates that the series did not converge. A minimum of 25 roots is used by the program.

For a solute that is not subject to first-order chemical transformation (X=0), equation 44 can be replaced (Clear-y and Adrian, 1973; Wexler and Clear-y, 1979) by

C(x,t)=C,

1

l-2 exp g-g [ 1 m f3iSiIl - exp -2

c

(y) ( Ppt)

. (46) i=l

For large values of time (st*eady-state solution), equation 44 can be reduced (van Genuchten and Alves, 1982, p. 58) to

C(x)=C,exp (47)

Linear equilibrium adsorption and ion exchange can be simulated by first dividing the coefficients D and V by the retardation factor, R (eq. 15). (Note: U in eqs. 44 and 47 would be given by U=dV*+4XD*). Tempo- ral variations in source concentration can be simulated through the principle of superposition (eq. 39).

Description of program FINITE

The program FINITE computes the analytical solution to the one-dimensional solute-transport equation for a finite system with a first-type (eq. 44) or third- type (eq. 52) source boundary condition at the inflow end. It consists of a main program and four subroutines-ROOTl, ROOT3, CNRMLl, and CNRML3. The function of the main program and subroutines ROOT1 and CNRMLl are outlined below; the program code listing is presented in attachment 2. Subroutines ROOT3 and CNRML3 are called when a third-type boundary condition is specified and are described in a subsequent section.

The program also calls the output subroutines TITLE, OFILE, and PLOTlD, which are common to most of the programs described in this report. These subroutines are described in detail later.

Main program

The main program reads and prints all input data needed to specify model variables. The required input data and the format used in preparing a data file are shown in table 1.

The program calls subroutine ROOT1 to compute the positive roots of equation 45 when a first-type source boundary condition is specified and then executes a set of nested loops. The inner loop calls subroutine CNRMLl to calculate the concentration for a particular time value and distance; the outer loop cycles through all specified time values and prints a table of concentration in relation to distance for each time value. Graphs of concentration in relation to distance can also be plotted.

Subroutines ROOT1 and CNRMLl

Subroutine ROOT1 calculates the roots of the equation

a*cot(a)+c=O

by an iterative procedure. The first root is known to lie between 1~/2 and 7~, and an initial estimate of 0.75 7~ is made. Newton’s second-order method (Salvadori and Baron, 1961, p. 6) is used to correct and update the estimate at each iteration. A maximum of 50 iterations and a convergence criterion of 1.0~10-‘~ are set in the subroutine. Each subsequent root of the equation is about r greater than the previous one. This value is used as an initial estimate in the search for the remaining roots.

Subroutine CNRMLl calculates the normalized concentration (C/C,,) for a particular time value and x-distance value, using equation 44 for a solute subject to first-order chemical transformation and equation 46 if the solute is conservative (h=O). The number of terms taken in the infinite series summation is specified in the input data.

Sample problems 1 a and 1 b

Two sample problems are presented that use data similar to the data given in Lai and Jurinak (1972). In sample problem la, a conservative solute is introduced into a saturated soil column under steady flow. Model variables are

Velocity (V) =0.6 in/h Longitudinal dispersion (D) =0.6 in21h System length (L) =12 in


Table I.-Input data format for the program FINITE

Data VEGAable set COlUUmS Format name DescriDtion

1 1 - 60 A60 TITLE Data to be printed in a title box on first page of program output. Last line in data set must have an -1" in column 1. First four lines are also used as title for plot. ------------------------------------------------------------------------------------------------------~-----

2 1-4 14 NBC Boundary condition type (NBC = 1 for a first-type boundary condition; NBC - 3 for a third-type boundary condition).

5-6 I4 Nx Number of x-coordinates at which solution will be evaluated.

S-12 14 NT Number of time values at which solution will be evaluated.

13 - 16 14 NROOT Number of terms used in infinite series sumnation.

17 - 20 14 IPLT Plot control variable. is greater than 0.

Concentration profiles will be plotted if IPLT ---------------------------------------------------------------------------------------~--------------------

3 1 - 10 A10 CUNITS Character variable used as label for units of concentration in program + output.

11 - 20 A10 VUNITS Units of ground-water velocity.

21 - 30 A10 DUNITS Units of dispersion coefficient.

31 - 40 A10 KUNITS Units of solute-decay coefficient.

41 - 50 .A10 LUNITS Units of length.

51 - 60 A10 TWITS Units of time. ----------------------------------------------------------------------------------------~------------------- 4 1 - 10 F1O.O CC Solute concentration at inflow boundary.

11 - 20 F10.0 VX Ground-water velocity in x-direction.l

21 - 30 F10.0 DX Longitudinal dispersion ooeffioient.1

31 - 40 F10.0 DK First-order solute-decay coefficient.'

41 - 50 F1O.O XL Length of flow system.'

51 - 60 F10.0 XSCLP Sealing factor by which x-coordinate values are divided to convert them to plotter inches. -----___--------------------------------------------------------------------------------~-------------------

5 1 - 80 8FlO.O X(I) X-;;zzfinates at which solution will be evaluated (elight values per -------_----------------------------------------------------------------------------------------------------

6 1 - 60 8FlO.O T(I) TimTn;;lues at which solution will be evaluated (ei6ht values per

'All units must be consistent.

Solute concentration at inflow boundary (C,)

=l.O mg/L. Retardation factor (R) =8.31 Scaled velocity (V’) =0.072 in/h

Concentrations are calculated for points 0.5 in apart at elapsed times of 2.5, 5, 10, 15, and 20 hours.

In sample problem lb, a solute is removed by linear equilibrium adsorption. Additional model variables are

Soil bulk density (pb) =0.047 lb(mass)/in3

Porosity (n) =0.45 Slope of adsorption isotherm (k) =70 in3Ab

(mass).

From these values and equations 15 and 16 (substitut- ing n for 0), the terms obtained are

Scaled dispersion coefficient (I)‘) =0.072 in2/h.

Concentrations are calculated for points 0.5 in apart at elapsed times of 20, 50, 100, and 150 hours.

The input data sets for sample problems la and lb are shown in figures 4A and 514; computer plots of concentration profiles generated by the program FINITE are shown in figures 4B and 5B. Comparison of the concentration profiles at 20 hours shows the retarding effect of adsorption on solute movement.

Program output for sample problem la is presented in attachment 4. Sample problems la and lb each required 3.9 seconds (s) of cen.tral processing unit (CPU) time on a Prime model 9955 Mod II.


A Sample Problem la -- Solute transport in a finite-length soil column with a first-type boundary condition at x=0 Model Parameters: L=12 in, FO.6 in/h, D=0.6 in**l/h

Kl=O.O per h, CO=l.O mg/L WC=

1 25 05 50 1 N/L IN/H IN"*2IH PER HOUR INCHES HOURS

1.0 E

0.6 0.0 12.0 1.2 0.0 4.0 415

El 2.0 2.5

1x*: 6.5 9:o k-z 9:5

6.0 6.5 10.0 10.5

2:5 5.0 10.0 15.0 20.0

3.0 3.5 7.0 7.5

11.0 11.5

f3

SampLe ProbLem lo -- Solute transport in o finite-Length soil column with o first-type boundary condition at x-0

FlodeL Parameters: L-12 in, V-O.6 in/h, O-O.6 inxx2/h Kl-0.0 per h, CO-l.0 mg/L

Figure 4.-(A) Sample input data set, and (B) concentration profiles generated by the program FINITE for a conservative solute in a finite-length system with first-type source boundary condition after 2.5, 5, IO, 15, and 20 hours (sample problem la).


A

B

Sample Problem lb -- Solute transport in a finite-length soil column with a first-type boundary condition at x=0 Model Parameters: L=12 in, V=O.O72 in/h. D=0.072 in**2/h

Kl=O.O per h, CO=l.O mg/L Solute is subject to linear adsorption ====

1 25 04 50 1 MG/L IN/H IN**P/H PER HOUR INCHES HOURS

1.0 0.072 0.072 0.0 12.0 0.0 0.5 1.0 1.5 2.0 z 4.0 4.5 5.0 5.5 6:5 8.0 8.5 9.0 9.5 10.5

12.0 20.0 50.0 100.0 150.0

3.0 7.0

11.0

3.5 7.5

11.5

1

0.9

0.6

0.7

5

E 0.6

E

E

z 0.5

z

2

2

0.4

z

0.3

0.2

0.1

0

T

0 1.2 2.4 3.6 4.6 6 7.2 8.4 9.6 10.6

Sample Problem lb -- Solute transport in a finite-Length soil column with a first-type baundory condition at x-0

Model Parameters: L-12 in, V-O.072 in/h, D-0.072 inrx2/h Kl-0.0 per h, CO-l.0 mg/L

-I

OISTRNCE RLONG X-AXIS, IN INCHES

Figure 5.-(A) Sample input data set, and (B) concentration profiles generated by the program FINITE for a solute subject to linear adsorption in a finite-length system with first-type source boundary condition after 20, 50, 100, and 150 hours (sample problem lb).

ANALYTICAL SOLUTIONS FOR SOLUTE TRANSPORT IN GROUND-WATER SYSTEMS WITH UNIFORM FLOW 1’7

a Finite system with third-type source boundary condition

Governing equation


E& -$E-,~ at ax2 ax

Boundary conditions:

VC =VC-DaC 0 ax' x=0

x=L

Initial condition:

c=o, O<x<L at t=O

Assumptions:

(48)

(49)

(50)

(51)

1.

0

2.

3. 4.

Fluid is of constant density and viscosity. Solute may be subject to first-order chemical transformation (for a conservative solute, h=O). Flow is in x-direction only, and velocity is constant. The longitudinal dispersion coefficient (D), which is equivalent to D, (eq. 7), is constant.

Analytical solution

The solution to equation 48 was first presented by Selim and Manse11 (1976). The following equation is modified from a form presented in van Genuchten and Alves (1982, p. 66-67):

C(x,t)=C,

I [ w+w- w-w2 -E

2v Bv(u+v) exp

( )I D

(52)

where U=dVL+4XD and pi are the roots of the equation

p cot pm+vL_o VL 4D ’ (53)

For a solute that is not subject to first-order chemical transformation (X=0), equation 52 can be simpli- tied (Gershon and Nir, 1969, p. 837; van Genuchten and Alves, 1982, p. 13) as

C(x,t)=C, 1 VL vx v2t l-2pxp 2D-m [ 1

(54)

For large values of time (steady-state solution), equation 52 can be reduced (van Genuchten and Alves, 1982, p. 59) to

C(x)=C,

{

exp~~]+~~~~~cxp[(~)-~] .(55)

[ (u+v) (u-v)2 - -2wJ+V~ex4-~l 2v I

Comments:

The roots of equation 53 can be found by standard root-search techniques. An iterative technique using Newton’s second-order correction method was described in the preceding section.

Linear equilibrium adsorption and ion exchange can be simulated by first dividing the coefficients D and V by the retardation factor, R (eq. 15). (Note: U in eqs. 52 and 55 would be given by U=dV*+4XD*). Tempo-


ral variations in source concentration can be simulated through the principle of superposition (eq. 39).

Description of program FINITE

The analytical solution to the one-dimensional solute-transport equation for a finite system with a third-type (or first-type) source boundary condition at the inflow end is computed by the program FINITE, described in detail in the preceding section. The main program reads and prints all input data needed to specify model variables. The required input data and the format used in preparing a data file are shown in table 1.

The main program then calls subroutine ROOT3 to compute the positive roots of equation 53 when a third-type source boundary condition is specified, and executes a set of nested loops. The inner loop calls subroutine CNRML3 to calculate the concentration for a particular time value and distance; the outer loop cycles through all specified time values and prints a table of concentration in relation to distance for each time value. Graphs of concentration in relation to distance can also be plotted.

Subroutines ROOT3 and CNRML3

Subroutine ROOT3 calculates the roots of the equation a-cot(a)-b*a’+c=O. The procedure followed is similar to that for subroutine ROOT1 (described in the preceding section), with ~12 as an initial estimate for the first root.

Subroutine CNRMLS calculates the normalized concentration (C/C,) for a particular time value and distance value, using equation 52 for a solute subject to first-order chemical transformation and equation 54 if the solute is conservative (X=0). The number of terms taken in the infinite series summation is specified in the input data.

Sample problem 2

In sample problem 2, the solute introduced into the soil column is assumed to be conservative. Model variables are identical to those in sample problem la and are

Velocity (V) =0.6 in/h Longitudinal dispersion (D) =0.6 in21h System length (L) =12 in Solute concentration opposite =l.O mg/L.

inflow boundary (C,)


The input data set for sample problem 2 is shown in figure 6A; a computer plot of concentration profiles generated by the program FINITE is shown in figure 6B. Output for this sample problem is presented in

attachment 4. Sample problem 2 required 4.3 s of CPU time on a Prime model 9955 Mod II.

Comparison of figures 4B and 6B shows that the principal difference between the solutions for a first- type and a third-type source boundary condition is reflected in the solute concentrations near the inflow boundary at early times. As mentioned previously, these differences decrease with decreasing values for the quantity D/V.

Semi-infinite system with first-type source boundary condition

Governing equation



c=c,, 2;=0

c ac,() tax 7 >; = cc

Initial condition:

c=o, o<x<a at t=O

Assumptions:

1. Fluid is of constant density and viscosity. -.

(57)

(5%

l (59) ’

2. Solute may be subject to first-order chemical transformation (for a conservative solute, X=0).

3. Flow is in x-direction only, and velocity is constant. 4. The longitudinal dispersion coefficient (D), which is

equivalent to D, (eq. 7), is constant.

(56)

Analytical solution

The following equation was modified from Bear (1972, p. 630) and van Genuchten and Alves (1982, p. 60):

+exp[&(V+U)]*erfc[ ;$I}, 030)

where U=dV”+4h.D.

The analytical solution for a solute not subject to first-order chemical transformation (X=0) was derived by Ogata and Banks (1961) as


A

Sample Problem 2 -- Solute transport in a finite-length soil column with a third-type boundary condition at x=0 Model Parameters: L=12 in, V=O.6 in/h, D=0.6 in**2/h

Kl=O.O per h, CO=l.O mg/L -

3 25 05 50 1 M/L IN/H IN**2/H PER HOUR INCHES HOURS

1.0 0.6 0.6 0.0 12.0 1.2 ::: 0.5 4.5 t*:

9:o

t.2

9:5

2.0 6.0 2.5 6.5 ?Z 3.5

1E 6.5 10.0 10.5 11:o 7.5

11.5

2:5 5.0 10.0 15.0 20.0

I

0.9

0.8

0.7

5

z 0.6 s 2 E

8 0.5

8 !z

2 E

0.4

B

0.3

0.2

0.1

0

SompLe ProbLem 2 -- SoLute transport in o finite-length soiL coLumn with o third-type boundary condition at x-0

Model Parameters: L-12 in, V-O.6 in/h, D-O.6 inwx2/h Kl-0.0 per h, CO-l.0 mg/L

0 1.2 2.4 OISTANCE FILONG 3.6 4.8 X-&S, IN7i:CtlES 8.4 9.6 10.6 12

igure 6.-(A) Sample input data set, and (B) concentration profiles generated by the program FINITE for a conservative solute in a finite-length system with third-type source boundary condition after 2.5, 5, 10, 15, and 20 hours (sample problem 2).


C,x,t,=$‘{erfc[~]+exp[#erfc[$]}. (61)

For large values of time (steady-state solution), equation 60 reduces (modified from Bear, 1972, p. 631) to

C(x)=C, exp &(V-U) [ 1 6%

Comments:

Equations 60 and 61 are presented in this form to utilize computer routines that accurately compute the product of an exponential term (exp [xl) and the complementary error function (denoted as erfc [y]).

Linear equilibrium adsorption and ion exchange can be simulated by first dividing the coefficients D and V by the retardation factor, R (eq. 15). (Note: U in eqs. 60 and 62 would be given by U=dV*+4XD*). Tempo- ral variations in source concentration can be simulated through the principle of superposition (eq. 39).

Description of program SEMINF

The program SEMINF computes the analytical solution to the one-dimensional solute-transport equation for a semi-infinite system with a first-type or third-type source boundary condition at the inflow end. It consists of a main program and two subroutines-CNRMLl and CNRML3. The function of the main program and subroutine CNRMLl are outlined below; the program code listing is presented in attachment 2. Subroutine CNRML3, called when a third-type boundary condition is specified, is described in a subsequent section.

The program also calls the subroutine EXERFC and the output subroutines TITLE, OFILE, and PLOTlD, which are common to most programs described in this report. These subroutines are described in detail later.

Main program


The program next executes a set of nested loops. The inner loop calls subroutine CNRMLl to calculate the concentration for a particular time value and distance. The outer loop cycles through all specified time values and prints a table of concentration in relation to distance for each time value. Graphs of concentration in relation to distance can also be plotted.

Subroutine CNRMLl

Subroutine CNRMLl calculates the normalized concentration (C/C,) for a particular time value and distance, using equation 60 for a solute subject to first-order chemical transformation and equation 61 if the solute is conservative (h=O).

Sample problems 3a and 3b

Two sample problems are presented. In sample problem 3a, a conservative solute is introduced into a long soil column. The system is idealized as being semi-infinite in length, with model variables as

Velocity (V) =0.6 in/h Longitudinal dispersion (D) =0.6 in”/h Solute concentration at inflow =l.O mg/L.

boundary (C,)


In sample problem 3b, solute is removed by both first-order solute decay and linear equilibrium adsorption. Additional model variables are

Solute half-life (T1,, ) =7.6 days Soil bulk density (pJ =0.047

lb(mass)/in3 Porosity (n) =0.45 Slope of adsorption isotherm (k) =70 in3/lb

(mass).

From these values, the following terms are obtained using equations 15 and 25:

Decay constant (X) =0.0038 per hour

Retardation factor (R) =8.31 Scaled velocity (V*) =0.072 in/h Scaled dispersion coefficient (D”) =0.072 in2/h.

Concentrations are calculated for points 0.5 in apart at elapsed times of 20, 50, 100, and 150 hours.

Input data sets for sample problems 3a and 3b are shown in figures 7A and 8A; computer plots of concentration profiles generated by the program SEM- INF are also shown. Output for sample problem 3a is presented in attachment 4. Sample problems 3a and 3b each required 3 s of CPU time on a Prime model 9955 Mod II.

Comparison of the concentration profiles at 20 hours in each plot (figs. 7B and 8B) shows the effect of both solute decay and adsorption on solute movement. Comparison of figures 7B and 48 shows the difference in concentration profiles that would result if the solution for a semi-infinite syst,em were used to simulate transport in a finite system. The most significant difference is the lower solute concentrations and the

-


Table 2 .-Input data format for the program SEMINF

Data VEUiable set Columns Format name DescriDtion

1 1 - 60 A60 TITLE Data to be printed in a title box on first page of program output. Last line in data set must have an *=" in colucm 1. First four lines are also used as title for plot. ________________--________c_____________--------------------------------------------------------------------

2 l-4 14 NBC Boundary condition type (NBC - 1 for first-type boundary condition; NBC - 3 for third-type boundary condition).

5-6 14 Nx t&saber of r-coordinates at which solution will be evaluated.

0 - 12 14 NT Xumber of time values at which solution will be evaluated.

13 - 16 14 IPLT Plot control variable. Concentration profiles will be plotted if IPLT is greater than 0. ________________________________________--------------------------------------------------------------------

3 1 - 10 A10 CUXITS Character variable used as label for units of concentration in program output.


21 - 30 A10 DUXITS Units of dispersion coefficient.

31 - 40 A10 XUNITS Units of solute-decay coefficient.

41 - 50 A10 LUXITS Units of length.

51 - 60 A10 TUXITS Units of time. --____---------_--__---------------------------------------------------------------------------------------- 4 1 - 10 F10.0 co Solute concentration at inflow boundary.

11 - 20 P10.0 vx Ground-water velocity in x-direction-l

21 - 30 F10.0 DX Longitudinal dispersion coefficient.'

31 - 40 F1O.O DK First-order solute decay coefficient.'

41 - 50 F10.0 XSCLP Scaling factor by which x-coordinate values are divided to convert them to plotter inches. ______----______--__----------------------------------------------------------------------------------------

5 1 - 00 6FlO.O X(I) X-~;;.inates at which solution will be evaluated (eight values per . ------------------__----------------------------------------------------------------------------------------

6 1 - 60 6FlO.O T(I) Time values at which solution will be evaluated (eight values per line).

'All units must be consistent.

steeper gradients near x=12.0 in (fig. 7B). As mentioned previously, differences between the two solutions decrease with increased column Peclet number (P) and lower values for the number of displaced pore volumes (T).

Semi-infinite system with third-type source boundary condition

Governing equation



VC,=VC+Dg, x=0

(63)

C aC

, dx=o, x=cc

Irzitial condition:

c=o, o<x<m at t=O

6’3

(66)

Assumptions:

1. Fluid is of constant density and viscosity. 2. Solute may be subject to first-order chemical trans-

formation (for a conservative solute, h=O). 3. Flow is in x-direction only, and velocity is constant. 4. The longitudinal dispersion coefficient (D), which is

equivalent to D, (eq. 7), is constant.

Analytical solution

The following equation is modified from Cleary and Ungs (1978, p. 10):


A Sample Problem 3a -- Solute transport in a semi-infinite soil column with a first-type boundary condition at x=0 Model Parameters: V=O.E in/h, D=0.6 in**2/h

Kl=O.O per h, CO=l.O mg/L

I MG 1 25 05 1

.--IL IN/H IN**2/H PER HOUR INCHES HOURS 1.0

i:; 0.6 0.0

0.0 1.0 1.5 ;.i 2:

12:o

ii:: Co" i:z 10.0 6:0 2.5 3.0 3.5

1::: 1;:: 11.5 7.5

2.5 5.0 10.0 15.0 20.0

Sample Problem 30 -- Solute transport in o semi-infinite s0i.L column with o first-type boundory condition ot x-0

ModeL Parameters: V-O.6 in/h, D-O.6 inHr2/h Kl-0.0 per h, CO-l.0 mg/L

Figure 7.-(A) Sample input data set, and (B) concentration profiles generated by the program SEMINF for a conservative solute in a semi-infinite system with first-type source boundary condition after 2.5, 5, 10, 15, and 20 hours (sample problem 3a).


A Sample Problem 3b -- Solute transport in a semi-infinite soil column with a first-type boundary condition at x=0 Model Parameters: V=O.O72 in/h, D=0.072 in**2/h

K1=0.0038 per h, CO=l.O mg/L Solute is subject to first-order decay and linear adsorption s==

1 25 04 1 m/L IN/H IN**P/A PER HOUR INCHES HOURS

0':: 0.072 0.5 0.072 1.0 0.0038 2; 2.5

12:o 2: 4.5 8.5 ;:: :*ii 9:5 8:0 8.5 10.0 10.5

20.0 50.0 100.0 150.0

Sample Problem 3b -- Solute transport in o semi-infinite soil column with a first-type boundary condition at x-0

Model Parameters: V-O.072 in/h, O-0.072 innw2/h Kl-0.0038 per h, CO-l.0 mg/L

- - - 0 I.2 2.4 3.6 4.8 6 7.2- L-4 9.c l0.i I2 DISTRNCE ALONG X-RXIS, IN INCHES

Figure 8.-(A) Sample input data set, and (B) concentration profiles generated by the program SEMINF for a solute subject to first-order decay and linear equilibrium adsorption in a semi-infinite system with first-type source boundary condition after 20, 50, 100, and 150 hours (sample problem 3b).


ax,t>= 4hD q2 exppg-At].erf~[~]

-(++l)exp[&,V+U)]*erfc[~]}, (67)

where

U=@+4hD.

For a conservative solute (h=O), the solution to equation 63 is given by Lindstrom and others (1967) and van Genuchten and Alves (1982, p. 10) as

C(x,t)=C,(ierfc[s]+ $exp[ -e]

-i[ I+$+g]expg)*erfc[s]). (68)

For large values of time (steady-state solution), equation 67 can be reduced (Gershon and Nir, 1969, p. 837) to

C(x)=C,(v+U)*exp 2D 2v [x,,-U)]. (69)

Comments:

Equations 67 and 68 are presented in this form to utilize computer routines that compute the product of an exponential term and the complementary error function. For extremely small values of X, calculations of concentration values using equation 67 may be subject to round-off errors as both the denominator in the first term and the terms within the bracket approach zero.

Linear equilibrium adsorption can be simulated by dividing the coefficients D and V by the retardation factor, R (eq. 15). Temporal variations in source concentration can be simulated through the principle of superposition (eq. 39).

Description of program SEMINF

The analytical solution to the one-dimensional solute-transport equation for a semi-infinite system with a third-type (or first-type) source boundary condition is computed by the program SEMINF, described in detail in the preceding section. The main program reads and prints all input data needed to

specify model variables. The required input data and the format used in preparing a data file are shown in table 2.

The program next executes a set of nested loops. The inner loop calls subroutine CNRML3 to calculate the concentration for a particular time value and distance. The outer loop cycles through all specified time values and prints a table of concentration in relation to distance for each time value. Graphs of concentration in relation to distance can also be plotted.

Subroutine CNRML3

Subroutine CNRML3 calculates the normalized concentration (C/C,> for a particular time value and distance, using equation 67 for a solute subject to first-order chemical transformation and equation 68 if the solute is conservative (X=0).

Sample problem 4

In sample problem 4, a conservative solute is introduced into a long soil column. The system is idealized as being semi-infinite in length, with model variables as

Velocity (V) Longitudinal dispersion (D) Solute concentration opposite inflow

=0.6 in/h =0.6 in2/h

boundary (C,) =l.O mg/L.

Concentrations are calculated for points spaced 0.5 in apart at elapsed times of 2.5, 5, 10, 15, and 20 hours.

The input data set for sample problem 4 is shown in figure 9A; a computer plot of concentration profiles generated by the program SEM INF is shown in figure 9B. Because of the third-type boundary condition, solute concentration computed near x=0 at early times differs from C,.

Program output for this sample problem is presented in attachment 4. Sample problem 4 required 3.6 s of CPU time on a Prime model 9955 Mod II.

Two-Dimensional Solute Transport

Several analytical solutions are available for the two-dimensional form of the solute-transport equation (eq. 10). These solutions can be used to simulate transport of contaminants from sources within relatively thin aquifers, provided t.he solute is generally well mixed throughout the thickness of the aquifer and vertical concentration gradients are negligible. Trans- port of contaminants within a vertical section along the centerline of a contaminant plume in a thick


B

Sample Problem 4 -- Solute transport in a semi-infinite soil column with a third-type boundary condition at x=0 Model Parameters: V=O.6 in/h, D=0.6 in**Z/h

Kl=O.O per h, CO=l.O mg/L ==c=

3 25 05 1 MG/i IN/H IN**Z/H PER HOUR INCHES HOURS

o’.i 4:o

0.6 0.5 0.6 1.0 0.0 1.5 t-i 2.5 4.5 5.0 5.5 6:0 6.5

8.0 8.5 9.0 9.5 10.0 10.5 12.0

2.5 5.0 10.0 15.0 20.0

3.0 7.0

11.0

3.5 7.5

11.5

Sompte Problem 4 -- Solute transport in 0 semi-infinite soil column with o third-type boundory condition ot x-0

Model Porometers: V-O.6 in/h, O-O.6 inn*2/h Kl-0.0 per h, CO-l.0 mg/L

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 1.2 2.4 3.6

OISTRNCE k:ONG X-F& IN7&HES 8.4 3.6 10.8 12

Figure 9.-(A) Sample input data set, and (B) concentration profiles generated by the program SEMINF for a conservative solute in a semi-infinite system with third-type source boundary condition after 2.5, 5, IO, 15, and 20 hours (sample problem 4).


aquifer can be simulated with these solutions if the solute source is wide enough that horizontal concentration gradients, which cause solute movement perpendicular to the centerline, are negligible.

In the first solution presented, the aquifer is assumed to be of infinite area1 extent and to have a continuous point source in the x,y plane (equivalent to a line source extending the entire thickness of the aquifer). Fluid having a known solute concentration is injected into the aquifer at a constant rate. It is further assumed that the injection rate is small, and that the uniform flowfield around the well is not disturbed. Solutions in which radial flow away from an injection well is considered are discussed by Hseih (1986). A solution for an area1 source where solute enters the aquifer at a known flux and concentration is given by Code11 and others (1982).

For the remaining solutions presented in this section, aquifers are assumed to be of semi-infinite length and to have a solute source at the inflow boundary (at x=0). The width of the aquifer can be treated as being finite or infinite in extent. In an infinite-width system, impermeable boundaries at the edges of the aquifer are presumed to be far enough away as to have a negligible effect on solute distribution within the area of interest. Idealized diagrams of both types of systems are shown in figure 10.

One type of source configuration, referred to as a “strip” source (Clear-y and Ungs, 1978), has a finite width extending from y=Y, to y=Y, at x=0 (fig. 10). The concentration within the strip is uniform and equal to C,. At the boundary of the strip source (at y=Y, or y=Y,), th e concentration is equal to 0.5 C,. Elsewhere along the inflow boundary, the concentration is zero. Combinations of strip sources could be used to simulate odd-shaped concentration distributions or multiple sources through use of the principle of superposition, as previously described.

A solute source can also have a “gaussian” concentration distribution (Cleary and Ungs, 19’78, p. 80) given by

[ 1 -(y-Y,>2 C=C,exp 2a2 , x=0, (70)

where C, =maximum concentration at center of gaussian

concentration distribution, Y, =y-coordinate of center of solute source (X,=0),

and (T =standard deviation of the gaussian distribu-

tion. A field situation in which a gaussian distribution can

be found is shown in figure 11. The solute concentration at the waste-disposal pond is unknown, but a line of monitoring wells downgradient from the site and

normal to the direction of flow s,hows a concentration distribution that approximates a gaussian curve. (This is expected, as the concentration distribution along a cross section normal to the direction of flow taken at any point downgradient from an ideal point source would be gaussian.) The standard deviation of the distribution can be determined from the data as

(Y-Y,) z!rZZzZ “d-2 ln(C/C,)’ (71)

where C is the concentration observed at a well a distance (y-Y,> away from the point of maximum concentration.

Solving equation 71 may lead to differing values of u if the observed data are not perfectly gaussian. An alternative procedure (R.M. Clear-y, Princeton Uni- versity, written commun., 1978) is to (1) normalize the data by dividing the observed concentrations by C,, (2) plot a histogram of the normalized concentration with respect to y, and (3) calculate the area under the curve. The standard deviation can be approximated by - a=area/V2n. A sample problem illustrating the use of both methods is presented later.

This section presents analytical solutions for an l Aquifer of infinite area1 extent with a continuous

point source, when fluid is injected at a constant rate and concentration,

l Semi-infinite aquifer of finite width with a strip source,

l Semi-infinite aquifer of infinite width with a strip source, and

l Semi-infinite aquifer of infinite width with a gaussian source.

All solutions can account for first-order solute decay. Four computer programs (POINT2, STRIPF, STRIP1 and GAUSS) were written to calculate concentrations in these systems as a function of distance and elapsed time.

Aquifer of infinite areal extent with continuous poinit source

Governing equidion

The analytical solution for a continuous point source has been presented by several authors, including Bear (1972, 1979), Fried (1975, p. 132), and Wilson and Miller (1978). The solution is derived by first solving the solute-transport equation for an instunt!neous point source and then integrating the solution over time. The two-dimensional solute-transport equation for an instantaneous point source is given by

-


dC --&=Dx~+Dy~-v~-Ac 3 Q' C,exp [ 1 w-x,>

c(x,Y>= 2Dx

Q' +-$ C$(x-X,>S(y-Y&t-t’) (72)


c !a() ‘ax ’

x= +aJ

c aCz() ‘ay ’ y= +w,

where V =V,, velocity in x-direction, Q’ =fluid injection rate per unit thickness of aqui-

fer, n =aquifer porosity,

dt =infinitesimal time interval, S =dirac delta (impulse) function,

X,,Y,=x- and y-coordinates of point source, and t’ =instant at which point source activates

(assumed to be 0).

Initial condition:

c=o, --co< y<+m and --co<x<+~ at t=O (75)

Assumptions:


formation (for a conservative solute, h=O). 3. Flow is in x-direction only, and velocity is constant

(no radial flow). 4. The longitudinal and transverse dispersion coeffi-

cients (D, and DJ are constant.

Analytical solution

The following equation, modified from Bear (1979, p. 274)) represents the analytical solution for an instantaneous point source integrated with respect to time, such that

C(x,y,t)= CoQ' oxp[ “(;; y Ii,‘; 4nrdD,D,

1 dT,(W

where T is a dummy variable of integration for the time integral.

The steady-state solution is given (modified from Bear, 1979, p. 274) as

ZnndD,D,

Kl\}, (77)

where K, is the modified Bessel function of second kind and zero order. Tables of values and polynomial approximations for K,(x) are given by Abramowitz and Stegun (1964, p. 37, p. 417422).

Comments:

The integral in equation 76 cannot be simplified further and must, therefore, be evaluated numerically. A Gauss-Legendre numerical integration technique, used in the computer program written to evaluate the analytical solution (eq. 76)) is described later.

The integral in equation 76 is difficult to evaluate correctly at x and y values near the point source. (Mathematically, when (x-X,> and (y-Y,> approach zero, the integral in eq. 76 becomes a form of the exponential integral, E,(t), which becomes infinite at t=O; see Abramowitz and Stegun, 1964, p. 228.) Farther away from the point source, generally when (x-XJ2 is larger than V2, a meaningful solution can be obtained.

Linear equilibrium adsorption and ion exchange can be simulated by first dividing Q’ and the coefficients D,, D,, and V by the retardation factor, R (eq. 15). Temporal variations in source concentration or multiple sources can be simulated through the principle of superposition.

Description of program POINT2

The program POINT2 computes the analytical solution to the two-dimensional solute-transport equation for an aquifer of infinite area1 extent with a continuous point source. It consists of a main program and the subroutine CNRMLB. The functions of the main program and subroutine are outlined below; the program code listing is presented in attachment 2.

The program also calls subroutine GLQPTS and the output subroutines TITLE, OFILE, PLOT2D, and CNTOUR, which are common to most programs described in this report. These subroutines are described in detail later.

Main program

The main program reads and prints all input data needed to specify model variables. The required input


A PLAN VIEW

A v \\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\\’ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ ,mpermeab,ebo”ndary \\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\,\\\\\\\\\

A

5

,’ ”

’ Zone of contamination

A’

-% \\ ?i- .\

VERTICAL SECTION A-A’

Land surface

.:I io& 6f contamination Direction of ground-water flow

--- I

e boundary .\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\~\\\

PLAN VIEW

_-


Zone of contamination

y=o x=0 Loo

Figure IO.-(A) Plan view and vertical section of idealized two-dimensional solute transport in an aquifer of semi-infinite length and finite width, and (B) plan view of idealized two-dimensional solute transport in an aquifer of semi-infinite length and infinite width.


I Direction of around-water flow

v=700 feet t 7 Monitoring wells

Zone of contamination y=200 feet

t

h\

y=o x= 06 x=0

100 I I I I ) 0 250 500 750 1,000

SOLUTE CONCENTRATION, IN MILLIGRAMS PER LITER

Figure Il.-(A) Plan view of a semi-infinite aquifer of infinite width showing location of waste-disposal pond and monitoring wells, and graph of (B) observed solute concentration values and gaussian curve used to approximate concentration distribution at x=0.


Table 3.-Input data format for the program POINT2

Data Variable set c01Lmms Format name Descriotion

1 1 - 60 A60 TITLE Data to be printed in a title box on the first page of program output. Last line in data set must have an *a" in column 1. First four lines are also used as title for plot. ------------------------------------------------------------------------------------------------------------

2 l- 4 14 Nx Number of x-coordinates at which solution will be evaluated.

5- 8 14 NY Number of y-coordinates at which solution will be evaluated.

9-12 14 NT Number of time values at which solution will be evaluated.

13 - 16 14 NMAX Number of terms used in the numerical integration technique (must be equal to 4, 20, 60, 104, or 2561.

17 - 20 14 IPLT Plot control variable. Contours of normalized concentration will be plotted if IPLT is greater than 0. ----------------------------------------------------------------------------------------~-------------------

3 1 - 10 A10 CUNITS Character variable used as label for units of concentration in program output.




41 - 50 A10 LIMITS Units of length.

51 - 60 A10 TUNITS Units of time. __________________----------------------------------------------------------------------~------------------- 4 1 - 10 F1O.O CO Solute concentration in injected fluid.

11 - 20 F10.0 vx Ground-water velocity in x-direction.

21 - 30 F10.0 DX Longitudinal dispersion coefficient.

31 - 40 F10.0 DY Transverse dispersion coefficient.

41 - 50 F1O.O DK First-order solute-decay coefficient. -------------------------------------------------------------------------------------------------------~----

5 1 - 10 F1O.O xc X-coordinate of point source.

11 - 20 F10.0 YC Y-coordinate of point source.

21 - 30 P10.0 w Fluid injection rate per unit thiclcness of aquifer'.

31 - 40 F1O.O KIN Aquifer porosity. -----------------------------------------------------------------------------------------~------------------ 6 1 - 80 8FlO.O X(I) X-coordinates at which solution will be evaluated (eight values per

line). -----------------------------------------------------------------------------------------~------------------

7 1 - 80 8FlO.O Y(I) Y-zcofinates at which solution will be evaluated (eight values per -----------------------------------------------------------------------------------------~------------------

8 1 - 60 8FlO.O T(I) Tim;n;lues at which solution will be evaluated (eight values per ------------------------------------------------------------------------------------------.------------------ 29 1 - 10 F1O.O XSCLP Scaling factor by which x-coordinate values are divided to convert them

to plotter inches.

11 - 20 F10.0 YSCLP Scaling factor used to convert y-coordinates into plotter inches.

21 - 30 F10.0 DELTA Contour increment for plot of normalized concentration (must be between 0.0 and 1.0).

'For the solution to be consistent, units of QM must be identical to those of the dispersion coefficients.

'Data line is needed only if IPLT (in data set 2) is greater than 0.

data and the format used in preparing a data file are The program next executes a set of three nested shown in table 3. The routine then calls the subroutine loops. The inner loop calls subroutine CNRMLB to GLQPTS, which reads the data file GLQ.PTS contain- calculate the concentration at all specified y- ing values of the positive roots and weighting func- coordinate values for a particular x-coordinate value tions used in the Gauss-Legendre numerical integra- and time. The middle loop cycles through all x- tion technique. coordinate values. The outer loop cycles through all


-

0

specified time values and prints a table of concentrations in relation to distance for each time value. Model output can also be plotted as a series of maps showing lines of equal solute concentration.

Subroutine CNRMLZ

Subroutine CNRMLZ calculates the normalized concentration (C/C,) for a particular time value and distance. The integral in equation 76 is evaluated through a Gauss-Legendre numerical integration technique. The Gauss integration formula used is given by Abramowitz and Stegun (1964) as

where

I1 f(x)dx=~wi l f(zi), -1 i=l

(78)

n =order of Legendre polynomial, wi =weighting functions,

f(z,) =value of integrand calculated with variable of integration equal to zi, and

zi =roots of nth order polynomial. The normalized roots of the Legendre polynomial and the corresponding weighting functions are passed by subroutine GLQPTS and scaled in the subroutine to account for the non-normalized limits of integration (from 0 to t rather than from - 1 to + 1).

The number of terms summed in the numerical integration (equivalent to the order of the polynomial) is specified by the user. Roots of the Legendre polynomial of order 4, 20, 60, 104, and 256 (from data in Cleary and Ungs, 1978) are provided in data file GLQ.PTS. In general, the more terms used in the integration, the more accurate the approximation; however, this must be weighed against the corresponding increase in computational effort and time. Additional discussions of the numerical integration technique are presented in a later section describing subroutine GLQPTS.

Sample problem 5

In sample problem 5, an abandoned borehole that penetrates a brackish artesian formation is discharg- ing into an overlying freshwater aquifer. Model variables are

Aquifer thickness =lOO ft Discharge rate = 1,250 ft3/d Ground-water velocity (V) =2 ftld Longitudinal dispersivity (a,) =30 ft Transverse dispersivity (aUt> =6 ft Source concentration (C,) =l,OOO mg/L Point-source location (X,,Y,> =o, 500 ft Aquifer porosity (n) =0.25.

From these values, the terms obtained are Discharge rate per unit thickness of aquifer (Q’) = 12.5 ft2/d

Coefficient of longitudinal dispersion (D,) =60 ft21d

Coefficient of transverse dispersion (D,) =12 ft2/d.

Concentrations are calculated at lo-ft intervals along the x-axis from x= -60 ft to x=200 ft, and at 5-ft intervals along the y-axis from y=450 ft to y=550 ft. Chloride concentration distribution after 25 days and 100 days is simulated.

The input data set for sample problem 5 is shown in figure 12A. A computer-generated contour plot of normalized concentrations (C/C,> at both time values is shown in figure 12B. Program output for this sample problem is presented in attachment 4. Sample problem 5 required 9 s of CPU time on a Prime model 9955 Mod II.

Aquifer of finite width with finite-width solute source

Governing equation

Two-dimensional solute-transport equation:

ac a2c dt- - Dx -+D,F-V $hC dX2 f

(79)


c=c,, x=0 and Y,< y<Y,

c=o, x=0 and y<Y, or y>Y, (80b)

c aC=, pay 7 y=o

c !a() )ay 9 y=w

c aC,o 'ax 9

x=cn

where V =velocity in x-direction,

(81)

(82)

(83)

Y, = y-coordinate of lower limit of solute source at x=0

Y, =y-coordinate of upper limit of solute source at x=0, and

W =aquifer width.


000 wr-.*

I r-4

PI 2 -q~ooo .

000 . 000 2s;‘:

m


-

a

Initial condition: Description of program STRIPF

c=o, O<x<m and O<y<W at t=O (84

Assumptions:


formation (for a conservative solute, h=O). 3. Flow is in x-direction only, and velocity is constant. 4. The longitudinal and transverse dispersion coeffi-

cients (D,, DJ are constant.

The program STRIPF computes the analytical solution to the two-dimensional solute-transport equation for an aquifer of finite width with a finite-width or “strip” solute source at the inflow boundary. It consists of a main program and subroutine CNRMLF. The functions of the main program and subroutine are outlined below; the program code listing is presented in attachment 2.

Analytical solution

The following equation is modified from Hewson

The program also calls the subroutine EXERFC and the output subroutines TITLE, OFILE, PLOTBD, and CNTOUR, which are common to most programs described in this report. These subroutines are described in detail later.

(1976):

C(x,y,t)=C,CL,P, cos (qy) n=O

l [ exp pp]erfc[ *]

+exp[p]erfc[ s]]

where

1 l/2, n=O

L= 1 , n>O

y2-Yl n=O

p,= W

i

[sin (qY,)-sin (qydl 00 n7r ,

q=n7FlW, n=0,1,2,3. . .

p=~V2+4D,(neD,+h)

Comments:

(85)

Terms in the infinite series in equation 85 tend to oscillate, and the series converges slowly for small values of x; thus, a large number of terms may be needed to ensure convergence. A good initial estimate is 100 terms. For larger values of x, the series converges more quickly.

The solution can yield results with either D, or X=0. Linear equilibrium adsorption and ion exchange can be simulated by first dividing the coefficients D,, D,, and V by the retardation factor, R (eq. 15). Temporal variations in solute concentration and odd-shaped source configurations can be simulated through the principle of superposition.

Main program


The program next executes a set of three nested loops. The inner loop calls subroutine CNRMLF to calculate the concentration at all specified y- coordinate values for a particular x-coordinate value and time. The middle loop cycles through all x- coordinate values. The outer loop cycles through all specified time values and prints a table of concentration in relation to distance for each time. Model output can also be plotted as a map showing lines of equal solute concentration.

Subroutine CNRMLF

Subroutine CNRMLF calculates the normalized concentration (C/C,) for a particular time value and distance using equation 85. The maximum number of terms in the infinite series summation is specified by the user. Because terms in the series tend to oscillate, a subtotal of the last 10 terms is kept, and when the subtotal is less than a convergence criterion set at 1 x 10-12, the series summation is halted. If the series does not converge after the specified maximum number of terms are taken, a warning message is printed on the program output.

Sample problem 6

In sample problem 6, migration of chloride ion in landfill leachate through a narrow, relatively thin, valley-fill aquifer is simulated. Model variables are

Aquifer width (W) Lower limit of solute source (YJ Upper limit of solute source (Y2> Ground-water velocity (V,) Longitudinal dispersivity (01~) Transverse dispersivity (at>

=3,000 ft =400 ft =2,000 ft =l ftld =200 ft =60 ft

34 TECHNIQUESOFWATER-RESOURCESINVESTIGATIONS

Table 4.-Input data format for the program STRIPF

Data V53Iiable set c01lmms Format name Description -

1 1 - 60 A60 TITLE Data to be printed in a title box on the first page of program output. Last line in data set must have au "=ll in colmm 1. First four lines are also used as title for plot. --_-_______---------____________________-------------------------------------------------~------------------

2 l-4 14 Nx Number of x-coordinates at which solution will be evaluated.

5-6 I4 NY Number of y-coordinates at which solution will be evaluated.

9 - 12 14 NT Number of time values at which solution will be evaluated.

13 - 16 14 NMAX Maximum number of terms used in the infinite series summation.

17 - 20 14 IPLT Plot control variable. Contours of normalized concentration will be plotted if IPLT is greater than 0. ----_______-------------------------------------------------------------------------------------------------





41 - 50 Al0 LUNITS Units of length.

51 - 60 A10 TUNITS Units of time. ------------------------------------------------------------------------------------------~--------------~-- 4 1 - 10 F10.0 co Solute concentration at inflow boundary.




41 - 50 F10.0 DK First-order solute-decay coefficient. ------------------------------------------------------------------------------------------~----------------- 5 1 - 10 F10.0 W Aquifer width (aquifer extends from y = 0 to y - W).

11 - 20 F10.0 Yl Y-coordinate of lower limit of finite-width solute source.

21 - 30 no.0 Y2 Y-coordinate of upper limit of finite-width solute source. -----------------------------------------------------------------------------------------~---------------~-- 6 1 - 60 6FlO.O X(I) X-coordinates at which solution will be evaluated (eight values per

line). ---________---------____________________--------------------------------------------------~----------------- 7 1 - 60 6FlO.O Y(I) Y-coordinates at which solution will be evaluated (eight values per

line). ____________---_________________________-------------------------------------------------------------------- 6 1 - 60 6FlO.O T(I) Tim;s;;lues at which solution will be evaluated (eight values per

___________------------------------------------------------------------------------------------------------- 19 1 - 10 F1O.O XSCLP Scaling factor by which x-coordinate values are divided to convert them

to plotter inches.

11 - 20 F1O.O YSCLP Scaling factor used to convert y-coordinates into plotter inches.

21 - 30 F1O.O DELTA Co;t.;wa!za!~~~t for plot of normalized concentration (must be between . .


Source concentration (C,) =l,OOO mg/L.

From these values, the terms obtained are

Dispersion in x-direction (D,) =200 fta/d Dispersion in y-direction (D,) = 60 ft2/d.

Concentrations are calculated at 150-ft intervals along the x-axis for 4,500 ft, and at 100% intervals along the y-axis for 3,000 ft. Chloride concentration distribution after 1,500 and 3,000 days is simulated.

The input data set for sample problem 6 is shown in figure 13A. A computer-generated contour plot of normalized concentration (C/C,) at each time value is shown in figure 13B. The lack of symmetry about the centerline of the chloride plume is due to the effect of the closer lateral boundary (at y=O). Lines of equal concentration are perpendicular to the lateral boundary, indicating that concentration gradients in the y-direction equal zero and, thus, no solute flux occurs across the boundary. Program output for this sample


000 000 . . . . . . RSt=: 888 ON4 FlcJm “22

00000000 . . . . . . . . 00000000 00000000 “Fl~~“~~c$

00000000

j- -u-l rica ou\o In- ‘*- E

a

133A NI 'SIX&A 3NOltl 33NHLSIO

133j NI 'SIX&A SNOW 33NUlSIO


problem is presented in attachment 4. Sample problem 6 required 52 s of CPU time on a Prime model 9955 Mod II.

A ?

uifer of infinite width with inite-width solute source

Governing equation


Bou?zdary conditions:

c=c,, x=0 and Y,<y<Y,

c=o, x=0 and y<Y, or y>Y,

(-J aC,() pay f y=+c.Q

(-J aC,(-J fax 7 x=w,

c3f3

(8%)

(87b)

c38)

(89)


Y, =y-coordinate of lower limit of solute source at x=0, and

Y, =y-coordinate of upper limit of solute source at x=0.

Initial condition:

c=o, O<X<=J and --co<y<+w at t=O (90)

Assumptions:

1. 2.

3. 4.

Fluid is of constant density and viscosity. Solute may be subject to first-order chemical transformation (for a conservative solute, h=O). Flow is in x-direction only, and velocity is constant. The longitudinal and transverse dispersion coefficients (D,, DJ are constant.

The program STRIP1 computes the analytical solution to the two-dimensional solute-transport equation for an aquifer of infinite width with a finite-width or “strip” solute source at the inflow boundary. It consists of a main program and the subroutine CNRMLI. The functions of the main program and subroutine are outlined below; the program code listing is presented in attachment 2.

The program also calls subroutines EXERFC and GLQPTS and the output subroutines TITLE, OFILE, PLOTZD, and CNTOUR, which are common to most programs described in this report. These subroutines are described in detail later.

Analytical solution Main program

The following equation is modified from Cleary and Ungs (1978, p. 17):

.{erfc[s]-erfc[z]}dT, @la>

To improve the accuracy of the numerical integration, a variable substitution can be made where 7=Z4, yielding

CCx,,N=-& ““I&] .tj$exp[ ~~(-g++--&]

l [e~c[~]-erfc[$$j+]}& (glb) Comments:

The integral in equation 91b cannot be simplified further and must be evaluated numerically. A Gauss- Legendre numerical integration technique was used in the computer program written to evaluate the analytical solution and is described later. Round-off errors may still occur when evaluating the solution for very small values of x at late times.

Linear equilibrium adsorption and ion exchange can be simulated by dividing the coefficients D,, D,, and V by the retardation factor, R (eq. 15). Temporal variations in solute concentration and odd-shaped source configurations can be simulated through the principle of superposition.

Description of progralm STRIPI

The main program reads and prints all input data needed to specify model variables. The required input data and the format used in preparing a data file are shown in table 5. The routine then calls the subroutine GLQPTS, which reads the data file GLQ.PTS containing values of the positive roots and weighting functions used in the Gauss-Legendre numerical integration technique.


Table 5.-Input data format for the program STRIPI

Data VEllZiElble set Columns Format name DescriDtiOn

1 1 - 60 A60 TITLE Data to be printed in a title box on the first page of program output. Last line in data set must have an W=ll in column 1. First four lines are also used as title for plot.

____________________---------------------------------------------------------------------------------------- 2 l-4 14 Nx Number of x-coordinates at which solution will be evaluated.

5-8 14 NY Number of y-coordinates at which solution will be evaluated.


13 - 16 14 NM4x Number of terms used in the numerical integration techniques (must be equal to 4, 20. 60. 104, or 2561.

17 - 20 14 IPLT Plot control variable. Contours of normalized concentration will be plotted if IPLT is greater than 0.

________--___---____------------------------------------------ ________________------------------------------ 3 1 - 10 A10 CUNITS Character variable used as label for units of concentration in program

output.




41 - 50 A10 LUNITS Units of length.

51 - 60 A10 TUNITS Units of time. __________-_________----------------------------------------------------------------------------------------

4 1 - 10 F1O.O co Solute concentration at inflow boundary.

11 - 20 F10.0 VX Ground-water velocity in x-direction.


31 - 40 F1O.O DY Transverse dispersion coefficient.

41 - 50 F1O.O DK First-order solute-decay coefficient. ________--__________----------------------------------------------------------------------------------- ----- 5 1 - 10 F1O.O Yl Y-coordinate of lower limit of finite-width solute source.

11 - 20 F10.0 Y2 Y-coordinate of upper limit of finite-width solute source. ________---_________----------------------------------------------- ----------_________----------------------

6 1 - 80 8FlO.O X(11 X-coordinates at which solution will be evaluated (eight values per line).

-----------_______--------------------------------------------- -----------________-------------------------- 7 1 - 80 8FlO.O Y(I) Y-coordinates at which solution will be evaluated (eight values per

line). ____________________----------------------------------------------------------------------------------------

8 1 - 80 8FlO.O T(1) Time values at which solution will be evaluated (eight values per line). _____________-______---------------------------------- ____________-_______----------------------------------

19 1 - 10 F10.0 XSCLP Scaling factor by which x-coordinate values are divided to convert them to plotter inches.


21- 30 F10.0 DELTA Contour increment for plot of normalized concentration (must be between 0.0 and 1.0).


The program next executes a set of three nested loops. The inner loop calls subroutine CNRMLI to calculate the concentration at all specified y- coordinate values for a particular x-coordinate value and time. The middle loop cycles through all x- coordinate values. The outer loop cycles through all specified time values and prints a table of concentration in relation to distance for each time. Model output can also be plotted as a map showing lines of equal solute concentration.

Subroutine CNRMLI

Subroutine CNRMLI calculates the normalized concentrations (C/C,,) for a particular time value and distance. The integral in equation 91 is evaluated through a Gauss-Legendre numerical integration technique. The normalized roots of the Legendre polynomial and the corresponding weighting functions are passed by subroutine GLQPTS and scaled in the subroutine to account for the non-normalized limits of integration (from 0 to t”4 rather than from - 1 to + 1).


The number of terms summed in the numerical integration (equivalent to the order of the polynomial) is specified by the user. Roots of the Legendre polynomial of order 4, 20, 60, 104, and 256 are provided in data file GLQ.PTS. In general, the more terms used in the integration, the more accurate the approximation; however, this must be weighed against the corresponding increase in computational effort and time. Additional discussions of the numerical integration technique are presented in a later section describing subroutine GLQPTS.

Sample problem 7

In sample problem ‘7, contaminant migration from a waste-disposal pond through the upper glacial aquifer of Long Island, N.Y., is simulated. Data are from a numerical modeling study by Pinder (1973). Model variables are

Lower limit of solute source (Y1> =635 ft Upper limit of solute source (Y.J =865 ft Ground-water velocity (V) =1.42 ftld Longitudinal dispersivity (0~~) =70 ft Transverse dispersivity (OLJ =14 ft Source concentration (C,) =40 mg/L.

Lateral boundaries are far enough from the area of interest that the aquifer can be treated as being infinite in width. From these values, the terms obtained are

Dispersion in x-direction (D,) =lOO fts/d Dispersion in y-direction (D,) = 20 ft’ld.

Concentrations are calculated at 100~ft intervals along the x-axis for 3,000 ft, and at 50-ft intervals on the y-axis for 1,500 ft. Concentration distributions after 5 years (1,826 days) are simulated.

The input data set for sample problem 7 is shown in figure 14A. A computer-generated contour plot of normalized concentration (C/C,> is shown in figure 14B. Program output for this sample problem is presented in attachment 4. Sample problem 7 required 1 min (minute) 25 s of CPU time on a Prime model 9955 Mod II.

Aquifer of infinite width with solute source having gaussian concentration

distribution Governing equation


(92)

Bounda y conditions:

-(Y-y,>2 C=C,exp 2a2 [ I , x=0 (93)

c aC,() 'ay 9 y=+oO (94

c ac,() 'ax 9

:<=cc, t (95)

where C, =maximum concentration at center of gaussian

solute source, Y, =y-coordinate of center of solute source at x = 0,

and (T =standard deviation of gaussian distribution.

Initial condition:

C=O, O<x<w and -c~yc:+~ at t=O (96)

Assumptions:


formation (for a conservative solute, h=O). 3. Flow is in x-direction only (V,=O), and velocity is

constant. 4. The longitudinal and transverse dispersion coeffi-

cients (D,, DY) are constant.

Arralytical solution

The following equation is and others (1980, p. 905):

modified from Gureghian

. (97) where

V2 P’4D,fA

and 7 is a dummy variable of integration for the time integral.

To improve the accuracy of the numerical integration, a variable substitution (modified from Cleary and Ungs, 1978, p. 20) can be made where 7=Z4, yielding


B

f

A

Sample Problem 7 -- Solute transport in a semi-infinite aquifer of infinite width with a continuous ‘strip’ source Model Data: V-l.42 ft/d, DX-100.0 ft**2/d, DY-20.0 ft**2/d

Yl-635 ft, Y2-865 ft, CO-40.0 mg/L

31 31 1 104 1 MG/L ET/D FT**2/D PER DAY FEET DAYS

40.0 1.42 100.0 20.0 0.0 635.0 865.0

0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 1100.0 1200.0 1300.0 1400.0 1500.0

1600.0 1700.0 1800.0 1900.0 2000.0 2100.0 2200.0 2300,O 2400.0 2500.0 2600.0 2700.0 2800.0 2900.0 3000.0

0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 400.0 450.0 500.0 550.0 600.0 650.0 700.0 750.0 800.0 850.0 900.0 950.0 1000.0 1050.0 1100.0 1150.0

1200.0 1250.0 1300.0 1350.0 1400.0 1450.0 1500.0 1826.0

500. 500. 0.1

Sam aqui P

Le Problem 7 -- Solute transport in a semi-infinite er of infinite width with a continuous ‘strip’ source

Model Data: V=1.42 ft/d, DX=lOO.O ft%c#2/d, DY=20.0 ft*G/d Y1=635 ft, Y2=865 ft, CO=40.0 mg/L

1 2000 I I I I I NORMALIZED CONCENTRATION AT TIME - 1826.DAYS CONTOUR INTERVAL - 0.1 C/Co

H J

U 0 500 1000 1500 2000 2500 3000


7gure 74.-(A) Sample input data set, and (B) computer plot of normalized concentration contours generated by the program STRIPI for a conservative solute in an aquifer of infinite width with finite-width solute source after 1,826 days (sample problem 7).


ZC,xa vx C(x,y,t)=~~~exp 2~,

[ I

q3z4w&k$f dZ

x 1 . , (98) where

Comments:

The integral in equation 98 cannot be simplified further and must be evaluated numerically. A Gauss- Legendre numerical integration technique was used in the computer program written to evaluate the analytical solution and is described later.

Linear equilibrium adsorption and ion exchange can be simulated by first dividing the coefficients D,, D,, and V by the retardation factor, R (eq. 15). Temporal variations in solute concentration can be simulated through the principle of superposition.

Description of program GAUSS

The program GAUSS computes the analytical solution to the two-dimensional solute-transport equation for an aquifer of infinite width with a solute source having a gaussian concentration distribution along the inflow boundary. It consists of a main program and the subroutine CNRMLG. The functions of the main program and subroutine are outlined below; the program code listing is presented in attachment 2.

The program also calls the subroutine GLQPTS and the output subroutines TITLE, OFILE, and PLOTBD, which are common to most programs described in this report. These subroutines are described in detail later.

Main program

The main program reads and prints all input data needed to specify model variables. The required input data and the format used in preparing a data file are shown in table 6. The routine then calls the subroutine GLQPTS, which reads the data file GLQ.PTS containing values of the positive roots and weighting functions used in the Gauss-Legendre numerical integration technique.

The program next executes a set of three nested loops. The inner loop calls subroutine CNRMLG to calculate the concentration at all specified y- coordinate values for a particular x-coordinate value and time. The middle loop cycles through all x-

coordinate values. The outer loop cycles through all specified time values and prints a table of concentration in relation to distance for each time value. Model output can also be plotted as a map showing lines of equal solute concentration.

Subroutine CNRMLC

Subroutine CNRMLG calculates the normalized concentration (C/C,) for a particular time value and distance. The integral in equation 98 is evaluated through a Gauss-Legendre numerical inte,gration technique. The normalized roots of the Legendre polynomial and the corresponding weighting functions are passed by subroutine GLQPTS and scaled in the subroutine to account for the non-normalized limits of integration, from 0 to t’” rather than from -1 to +l.

The number of terms summed in the the numerical integration (equivalent to the order of the polynomial) is specified by the user. Roots of the Legendre polynomial of order 4, 20, 60, 104, and 256 are provided in data file GLQ.PTS. In general, the more terms used in the integration, the more accurate the approximation; however, this must be weighed against the corresponding increase in computational effort and time. Additional discussions of the numerical integration technique are presented in a later section describing subroutine GLQPTS.

Sample problems 8a and 8b

Two sample problems are presented. Sample problem 8a is modified from an example presented in Gureghian and others (1980) for a conservative solute uniformly mixed in a thin aquifer of infinite width. Model variables are

Maximum concentration (C,) =l,OOO mg/L Standard deviation of gaussian

distribution (a) =130 ft Center of solute source (Y,) =450 ft Ground-water velocity (V,) =4 ftkl Coefficient of longitudinal

dispersion (D,) =150 ft2/d Coefficient of transverse dispersion

@,I =30 ft”/d. Concentrations are calculated at 50-ft intervals along the x-axis for 1,700 ft, and at 25ft intervals on the y-axis for 900 ft. The chloride concentration distribution after 300 days is simulated.

Sample problem 8b demonstrates two methods of calculating a value for u. Aquifer dimensions, ground- water velocity, and dispersion coefficients are the same as in problem 8a. Concentrations measured in monitoring wells 500 ft downgradient from a waste- disposal site are presented in table 7; figure 15 presents a plot of the normalized concentration (C/C,> in


Table 6.-Input data format for the program GAUSS

Data VS3dable set COlWrmS Format name DescriDtion

1 1 - 60 A60 TITLE Data to be printed in a title box on the first page of program output. Last line in data set must have an "=" in column 1. First four lines are also used as title for plot. ______----__________----------------------------------------- ------------____-__----------------------------




13 - 16 14 NnAx Number of terms used in the numerical integration technique (must be equal to 4, 20. 60, 104. or 256).

17 - 20 14 IPLT Plot control variable. Contours of normalized concentration will be plotted if IPLT is greater than 0. ________________________________________--------------------------------------------------------------------






51 - 60 A10 TUNITS Units of time. -__--__----_________---------------------------------------------------------------------------------------- 4 1 - 10 F1O.O a4 Maximum solute concentration at inflow boundary.




41 - 50 F1O.O DK First-order solute-decay coefficient. ----------_------------------------------------------------ ------------_______------------------------------

5 1 - 10 F10.0 YC Y-coordinate of center of gaussian-distributed solute source.

11 - 20 F10.0 ws Standard deviation of gaussian distribution describing solute source. ____________--______------------------------------------ ---_---------____----------------------------------- 6 1 - 60 6FlO.O X(I) X-coordinates at which solution will be evaluated (eight values per

line). ____________--______----------------------------------- ____________________---------------------------------

7 1 - 60 8FlO.O Y(I) Y-coordinates at which solution will be evaluated (eight values per line).

_-___---_-__________---------------------------------------- --__--------___-_------------------------------- 8 1 - 80 8FlO.O T(I) Time values at which solution will be evaluated (eight values per

line). ___________--_______--------------------------------- __________----______----------------------------------- 19 1 - 10 F10.0 XSCLP Scaling factor by which x-coordinate values are divided to convert them

to plotter inches.


21 - 30 F10.0 DELTA Co;t"XufFyyt for plot of normalized concentration (must be between . .


relation to distance along the y-axis (normal to the Input data sets for sample problems 8a and 8b are direction of flow). An average value of u, calculated shown in figures 16A and 17A. Computer-generated

l from the observed concentrations (table 7) using equa- contour plots of normalized concentration (C/C,) are tion 70, is 66.1 ft. The area under the curve in figure shown in figures 16B and 17B. Comparison of figures 15 can also be approximated and yields a u value of 16B and 17B shows the effect of varying o on the 65.0 ft. A value of 65 ft was used in the input data for concentration distribution. Program output for sample sample problem 8b. problem 8a is presented in attachment 4. Sample


Table 7.-Measured solute concentrations in monitoring wells downgradient from the waste-disposal site in sample problem 8b

well locations shown in fig. 111

we11 location (x and y

coordinates), in feet

0, 200

0, 250

0, 300

0, 350

0, 400

0, 450

0, 500

0, 550

0, 600

0, 650

0, 700

Measured solute Calculated value concentration, in of 0, in feet

milligrams per liter (from eq. 71)

2 70.9

12 67.2

65 64.2

310 65.3

725 62.3

1,000 __

760 67.5

290 63.6

82 67.1

9 65.2

1 67.3

problems 8a and 8b required 24 s of CPU time on a Prime model 9955 Mod II.

Three-Dimensional Solute Transport

Several analytical solutions are available for the three-dimensional form of the solute-transport equation (eq. 9), including those presented in Cleary and Ungs (1978), Huyakorn and others (1987), Code11 and others (1982>, Sagar (1982), and Hunt (1978). These solutions are particularly useful, as they can simulate transport of contaminants from sources in relatively thick aquifers when both vertical and horizontal spread of the solute is of interest. In addition to a solution modified from Cleary and Ungs (1978, p. 24-25), two solutions were derived by the author for this report. Detailed derivations of these solutions are presented in attachment 1.

In the first solution presented, the aquifer is assumed to be of infinite extent along all three coordinate axes. Fluid is injected into the aquifer through a point source at a constant rate and solute concentration (C,). It is further assumed that the rate of injection is low and does not disturb the predomi- nantly uniform flow field. In the remaining solutions presented in this section, the aquifer is assumed to be semi-infinite in length and to have a solute source located along the inflow boundary. The semi-infinite aquifer can be either finite in both width and height, extending from y=O to y=W and from z=O (the base

of the aquifer) to z=H, or infinite in width and height. A diagram of an idealized three-dimensional aquifer of semi-infinite length and finite width and height is presented in figure 18.

The solute source, referred to as a “patch” source (Cleary and Ungs, 1978), is of finite width and height and extends from y=Y, to y=Y, and from z=Z, to z=Z, at x=0 (fig. 18). The concentration within the patch is uniform and is equal to C,, except along the boundary of the patch source, where it is equal to 0.5 C,. Elsewhere along the inflow boundary, the concentration is 0. Combinations of patch sources could be used to simulate odd-shaped concentration distributions or multiple sources through the principle of superposition. First-order solute decay, adsorption, and ion exchange can also be simulated. A solution for a “gaussian source” of finite height along the boundary is given in Huyakorn and others (1987).

Three computer programs, POINT3, PATCHF, and PATCHI, were developed to calculate concentrations in these systems as a function of distance and elapsed time. They are described in this section.

Aquifer of infinite extent with continuous point source

Governing equation

The analytical solution for a continuous point source has been derived by first solving the solute-transport equation for an instantaneous point source and then integrating the solution over time. The three- dimensional solute-transport equation for an instantaneous point source is given by

.6(x-X,)6(y-Y,)6(z-Z,)6(t -tr>. (99)


c aC,() ‘ax ’

X”_foO

c aC,() ‘ay ’ y==-

c CL, ‘az ’

z= +m f

where V =velocity in x-direction, Q =fluid injection rate, dt =infinitesimal time interval,

S( > =dirac delta function,

(101)

(102)


0.8

p 2 ii ’ 0.8

8

0.2

0 100 200 300 400 yc 500 600 700

DISTANCE ALONG’ Y-AXIS, IN FEET

EXPLANATION

g =STANDARD DEVIATION OF THE GAUSSIAN DISTRIBUTION

yc =Y COORDINATE OF CENTER OF SOLUTE SOURCE

Histogram of normalized observed concentration values

Normalized observ concentration vah

red Je I

I A--

/

\ _ Area beneath histogram=183 feet r

Gaussian curve used to approximate source

concentration distribution

Figure 15.-Normalized concentrations in relation to distance for the waste-disposal site in sample problem 8a and fitted gaussian distribution.

X,,Y,,Z, =coordinates of point source, and t’ =time at which instantaneous point source

activates (assumed t,o be 0).

Initial Condition:

C=O, -w<x<a~, -w<y<a~, --cc,<z<w at t’=O (103)

Assumptions:


formation (for a conservative solute, X=0>. 3. Flow is in x-direction only, and velocity is constant.

This presumes that the fluid injection rate is small

and that the spread of solute due to radially diverging flow paths is negligible.

4. The coefficients of longitudinal dispersion (D,) and transverse dispersion (D,, D,), from equation 7, are constant.

Analytical solution

Hunt (1978, p. 76) presented a solution for a point source with a conservative solute. A solution for the instantaneous point source with solute decay was derived by the author using exponential Fourier transforms (detailed derivation in attachment 1) and can be expressed as

44

R

TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS

A - -- Sample Problem 8a -- Solute transport in.s semi-infinite aquifer of infinite width with a continuous gaussian source Model Data: V-4.0 ft/d, DX-150.0 ft**2/d, DY-30.0 ft**2/d

us-130 ft, YC-450 ft. co-1000.0 mg/L --cw

33 37 1104 1 MG/L FT/D FT**2/D PER DAY FEET DAYS

1000.0 4.00 150.0 30.0 0.0 450.0 130.0

0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 400.0 450.0 500.0 550.0 600.0 650.0 700.0 750.0 800.0 850.0 900.0 950.0 1000.0 1050.0 1100.0 1150.0

1200.0 1250.0 1300.0 1350.0 1400.0 1450.0 1500.0 1550.0 1600.0

0.0 25.0 50.0 75.0 100.0 125.0 150.0 175.0 200.0 225.0 250.0 275.0 300.0 325.0 350.0 375.0 400.0 425.0 450.0 475.0 500.0 525.0 550.0 575.0 600.0 625.0 650.0 675.0 700.0 725.0 750.0 775.0 800.0 825.0 850.0 875.0 900.0 300.0 250.0 250.0 0.1

Sam 7

le Problem 80 -- Solute transport in a semi-infinite aqua er of infinite width with a continuous aussion source ModeL Data: V-4.0 ft/d, DX-150.0 ftxx2/d, O%-30.0 ftxx>!/d

1250 I WS-130 ft, YC-450 ft, CO-1000.0 mg/L

-- NORMdLIZED COkENTRATIdN AT TIME'- 300.0dYS ' CONTOUR INTERVAL - O.lC/Co

250

0 ' I I I 1 I I --I 0 250 500 750 1000 1250 1500 1750


Figure 16.-(A) Sample input data set, and (B) normalized concentration contours generated by the program GAUSS for a conservative solute in an aquifer of infinite width having a gaussian concentration distribution (a=150 feet) at the inflow boundary at 300 days (sample problem 8a).


A I

Sample Problem 8b -- Solute transport in a semi-infinite aquifer of infinite width with,a continuous gaussian source Model Data: V-4.0 ft/d, DX-150.0 ft**2/d, DY-30.0 ft**2/d

ws-65 ft. YC-450 ft, co-1000.0 lug/L ----

33 37 1 104 1 MG/L FT/D FT**Z/D PER DAY FEET DAYS

1000.0 4.00 150.0 30.0 0.0 45070

0.0 400.0 800.0

1200.0 1600.0

0.0 200.0 400.0 600.0 800.0 300.0 250.0

65.0 50.0

450.0 850.0

1250.0

100.0 150.0 200.0 250.0 300.0 500.0 550.0 600.0 650.0 700.0 900.0 950.0 1000.0 1050.0 1100.0

1300.0 1350.0 1400.0 1450.0 1500.0

25.0 50.0 75.0 100.0 225.0 250.0 275.0 300.0 425.0 450.0 475 .O 500.0 625.0 650.0 675.0 700.0 825.0 850.0 875.0 900.0

250.0 0.1

350.0 750.0

1150.0 1550.0

125.0 150.0 175.0 325.0 350.0 375.0 525.0 550.0 575.0 725.0 750.0 775.0

B

Sam 7

le Problem 8b -- Solute transport in a semi-infinite oqu~ er of infinite width with o continuous oussian source ModeL Data: V-4.0 ft/d, 0X-150.0 ftxx2/d, 0 -30.0 ftux2/d 3(

WS-65 ft, YC-450 ft., CO-1000.0 mg/L 1250

I------ '

I I I I NORMALIZED CONCENTRATION AT TIME - 3OO.DRYS CONTOUR INTERVRL - O.lC/Co

0 I I I 1 I I

0 250 500 750 1000 1250 1500 1750 DISTANCE ALONG X-AXIS, IN FEET

Figure 17.-(A) Sample input data set, and (B) normalized concentration contours generated by the program GAUSS for a conservative solute in an aquifer of infinite width having a gaussian concentration distribution (a=65 feet) at the inflow boundary at 300 days (sample problem 8b).


PLAN VIEW

t

Y

\\\\\\\\ y=w \\\\\A\\\\\\\\\\, Impermeable boundary \\\\\k\\\\\\\\\\\\\\\‘.\\ \R\\\\\\.,,\\\\\\\\\\\\\\\\,,,,,.,,,,,,,,,,,\\\\\\\\\\\\\\\\\\\\\~

Zone of contamination


VERTICAL SECTION A-A’ A

JI \Z

z=h

Land surface c) \b”

V Water table

Zone of contamination Direction of

ground-water flow

Figure 18.-Plan view and vertical section of idealized three-dimensional transport in an aquifer of semi-infinite length and finite width and height.


l C(x,y,z,t)= c,Qdt exp p&fqg+A)(,-tY] 8na 3/2 (t-t’) 3/2 ~D,D,D,

‘exp (x-x,)2 (Y-Y,)2 @-ZJ2

-4D,(t-t’)-QD,(t-t’)-4D,(t-t’) * (lo4) 1 Equation 104 can be integrated with respect to time

to yield a closed-form solution for the continuous solute source as

C(x,y,z,t)=

+exp[ z]erk[ s] ),

where

D,(y-Y,)‘+D,(z-ZJ2 l/2 D

Y DZ 1

(105)

When X=0, equation 105 reduces to a form similar to that presented in Hunt (1978, p. 77) for a continuous point source with a conservative solute.

Comments:

Equation 105 is valid only when y does not equal zero. Also, concentrations determined at locations close to the point source may exceed C, for certain combinations of values of Q, V, D,, D,, and D,. In general, this can occur when Q is large relative to

n’D,D,.

A solution that accounts for radial flow away from the point source would be more appropriate at large injection rates.

Linear equilibrium adsorption and ion exchange can be simulated by dividing the coefficients Q, V, D,, D,, and D, by the retardation factor, R (eq. 15). Temporal variations in solute concentration can be simulated through the principle of superposition.

Description of program POINT3

The program POINT3 computes the analytical solution to the three-dimensional solute-transport equa-

tion for an aquifer of infinite extent with a continuous point source. It consists of a main program and the subroutine CNRML3. The functions of the main program and subroutine are outlined below; the program code listing is presented in attachment 2.

The program also calls the subroutine EXERFC and the output subroutines TITLE, OFILE, PLOT3, and CNTOUR, which are common to most programs described in this report. These subroutines are described in detail later.

Main program


The program next executes a set of four nested loops. The innermost loop calls subroutine CNRML3 to calculate the concentration at all specified y- coordinate values for a particular x-coordinate value, z-coordinate value, and time. The second loop cycles through all x-coordinate values. The third loop cycles through all z-coordinate values and prints a table of concentration in relation to x and y for each z value. The outer loop cycles through all specified time values. Model output can be plotted as a series of maps showing lines of equal solute concentration in a horizontal (x-y plane) cross section at each point along the z-axis.

Subroutine CNRML3

Subroutine CNRMLS calculates the normalized concentration (C/C,> for a particular time value and distance using equation 105. A warning message is printed on the program output if the values of (x-X,), (y-Y,>, and (z-Z,> all equal to zero are passed to the subroutine.

Sample problem 9

In sample problem 9, a natural gradient tracer test was conducted by injecting a chloride solution into an aquifer. The solution was injected through three wells spaced 2 ft apart, laterally, each having a small screened interval centered about z=lO ft. A total of 22.5 gallons (3 ft3) of solution was injected during a 24-hour period. Other model variables are

Aquifer porosity (n) =0.25 Ground-water velocity (V,) =O.l ftld Longitudinal dispersivity ((YJ =0.60 ft Horizontal transverse dispersivity (tit,,) =0.03 ft Vertical transverse dispersivity (a,,> =0.006 ft Chloride concentration in injected

solution =l,OOO mg/L


Table B.-Input data format for the program POINT3

Data Variable set Columns Format name Descriution

1 1 - 60 A60 TITLE Data to be printed in a title box on the first page of program output. Last line in data set must have an "=" in column 1. First four lines are also used as title for plot. ------------------------------------------------------------------------------------------------------~-----


!i- 6 14 NY Number of y-coordinates at which solution will be evaluated.

9-12 14 NZ Number of z-coordinates at which solution will be evaluated.








51 - 60 A10 QUNITS Units of solution injection rate.

61 - 70 A10 TUNITS Units of time.

4 1 - 10 F10.0 CO Solute concentration in injected fluid.

11 - 20 F10.0 VX Ground-water velocity in x-direction.


31 - 40 F10.0 DY Transverse dispersion coefficient in y-direction.

41 - 50 F10.0 DZ Transverse dispersion coefficient in z-direction.

__-____I'_I_"9___"9Ip_____DK___________--------------------------------~---------------------------------- First-order solute-decay coefficient

5 1 - 10 F10.0 XC X-coordinate of continuous point source.

11 - 20 F10.0 YC Y-coordinate of continuous point source.

21 - 30 F10.0 2c Z-coordinate of continuous point source.

31 - 40 F10.0 QM Solution injection rate.

41 - 50 F1O.O FOR Aquifer porosity.

6 1 - 60 6FlO.O X(I) X-coordinates at which solution will be evaluated (eight values per line).

----------------------------------------------------------------------------------------.-------------------- 7 1 - 80 6FlO.O Y(I) Y-;y;zinates at which solution will be evaluated (eight values per

______________-__-__--------------------------------------------------------------------~------------------- 6 1 - 80 8FlO.O Z(I) Z-coordinates at which solution will be evaluated (eight values per

line). ________________________________________------------------------------------------------~------------------- 9 1 - 80 6FlO.O T(I) Time values at which solution will be evaluated (eight values per

line). ________________________________________------------------------------------------------~------------------- 110 1 - 10 F10.0 XSCLP Scaling factor by which x-coordinate values are divided to convert them

to plotter inches.


21 - 30 F1O.O DELTA Co;t;wmpc,rTt for plot of nonaalised concentration (must be between . .



Injection well coordinates (X,, Y,, Z,) Well 1 =(O, 98, 10) Well 2 =(O, 100, 10) Well 3 =(O, 102, 10).


Coefficient of longitudinal dispersion W =0.06 f&d

Coefficient of horizontal transverse dispersion (DJ = 0.003 fta/d

Coefficient of vertical transverse dispersion (D,) = 0.0006 f&d

Injection rate per well (Q,) =l.O ft31d.

Chloride concentrations are computed in the z=lO- ft plane at x=0 and at 2-ft intervals along the x-axis from x=20 ft to x=60 ft, and at 1-ft intervals along the y-axis from y=90 ft to y=llO ft, after an elapsed time of 400 days. The injection period was simulated using the principle of superposition by first calculating the concentrations resulting from a continuous point source after 400 days and then subtracting the concentrations resulting from a continuous point source after 399 days. The effect of the multiple injection wells was simulated by summing the calculated concentrations for each well.

Rather than running the program POINT3 six times and then summing all the concentration values manually, it was easier to temporarily modify the main program by adding nine lines within the innermost loop, as follows:

DO 50 IY=l,NY YY=Y(IY)-YC CALL CNRML3(QM,POR,DK,T(IT),XX,YY,ZZ,

DX,DY,DZ,VX,CN) CXY(IX,IY)=CO*CN

YYl=YY +2.0 YY2=YY-2.0 CALL CNRML3(QM,POR,DK,T(IT),XX,YYl,

ZZ,DX,DY,DZ,VX,CNl) CALL CNRML3(QM,POR,DK,T(IT),XX,YY2,

ZZ,DX,DY,DZ,VX,CN2) Tl=T(IT)-1.0 CALL CNRML3(QM,POR,DK,Tl,XX,YY,ZZ,

DX,DY,DZ,VX,CN3) CALL CNRML3(QM,POR,DX,Tl,XX,YYl,ZZ,

DX,DY,DZ,VX,CN4) CALL CNRML3(QM,POR,DK,Tl,XX,YY2,ZZ,

DX,DY,DZ,VX,CN5) CXY (IX, IY) = CXY (IX, IY)+ CO* (CNl + CN2

-CN3-CN4-CN5) 50 CONTINUE

The input data set for sample problem 9 is shown in figure 19A; a computer-generated contour plot of normalized concentrations (C/C,) in the x-y plane at z=lO ft are shown in figure 19B. Sample problem 9 required 5 s of CPU time on a Prime model 9955 Mod II.

Aquifer of finite width and height with finite-width and finite-height

solute source

Governing equation

Three-dimensional solute-transport equation:

g+D$+D a2c+D$-V~-kC (106) “af


c=c,, x=0 and Y,<y<Y, and Z,<z<Z,

(107a)

c=o, x=0 and y<Y, or y>Y, and z<Z, or z>Z,

(107b)

c aC,o 'ay 9 y=o (108)

c a-(-J 'ay 9 y=w (109)

c aC,() faz 7 z=o (110)

c aC,o faz 7 z=H (111)

c aC,() 'ax 7 x=ca (112)


Y, =y-coordinate of lower limit of solute source, Y, =y-coordinate of upper limit of solute source, Z, =z-coordinate of lower limit of solute source, Z, =z-coordinate of upper limit of solute source at

x=0, W =aquifer width, and H =aquifer height.

Initial condition:

C=O O<x< 00, O<y<W, and O<z< H at t=O (113)

Assumptions:

1. Fluid is of constant density and viscosity.


A I

Sample Problem 9 -- Solute transport in an infinite aquifer with multiple point sources of finite duration Model Data: V-O.1 ft/d, DX-0.06 ft**2/d, DY-0.003 ft**2/d

DZ-0.0006 ft**2/d, QM-1.0 ft**3/d, CO-1000.0 mg/L, n-O.25 ---

21 21 1 01 1 MG/L FT/D FT**Z/D PER DAY FEET FT**3/D DAYS

1000.0 0.1 0.06 0.003 0.0006 0.0 100.0 10.0 1.00 0.25

20.0 22.0 24.0 26.0 28.0 30.0 32.0 34.0 36.0 38.0 40.0 42.0 44.0 46.0 48.0 50.0 52.0 54.0 56.0 58.0 60.0 90.0 91.0 92.0 93.0 94.0 95.0 96.0 97.0 98.0 99.0 100.0 101.0 102.0 103.0 104.0 105.0

106.0 107.0 108.0 109.0 110.0 10.0

400.0 5.0 5.0 0.01

Sample Problem 9 -- Solute transport in an infinite aquifer with muLtipLe point sources of finite duration

Model Doto: v=O.l ft/d, 0X=0.06 ftHn2/d, DY=0.003 ftnx2/‘d OZ-0.0006 ftxn2/d, ON-l.0 ftnn3/d, CO=lOOO.O mg/L, n=O.25

115 I I 1 I I I I NORMALIZED CONCENTRRTIONR;; ;+M; 1 ;';O;Fd;"

CONTOUR INTERVAL - O.OlC/Co

110 - E It z

90 I I I I I I I 20 25

;;STANCE :LONG X-&S, IN +&ET 50 55 60

Figure 19.-(A) Sample input data set, and (B) normalized concentration contours generated by the program POINT3 for a natural gradient tracer test in an aquifer of infinite extent after 400 days in the z=lO-foot plane (sample problem 9).


2.

a 3. 4.

Solute may be subject to first-order chemical transformation (for a conservative solute, X=0). Flow is in x-direction only, and velocity is constant. The coefficients of longitudinal dispersion (D,) and transverse dispersion (D,, D,), from equation 7, are constant.

Analytical solution

The solution to equation 106 was first derived by Cleary and Ungs (19’78, p. 24-25). A modified form of the equation (derived in detail by the author in attachment 1) can be given as

m cc

w,Y,w=Co~ ~h-rl,o,P, cm (54 cos (qy) m=O n=O

l { exp pF]*erfc[ s]

(114)

where

m=O, and n=O m=O, and n>O m>O, and n=O m>O, and n>O

1 ;J,-z,

H O,=

[sin ({Z,)-sin (&)I m7r

i

Y,--Yl Pn= W

[sin (qY,)-sin (7jYJl

m=O

m>O

n=O

n>O

t=m7r/H m=0,1,2,3. . .

q=n7rlW n=0,1,2,3. . .

P=~V”+4D,(rlZD,+5”D,+X).

Comments:

The terms in the infinite series in equation 114 tend

0 to oscillate, and the double series converges slowly for small values of x and time. Therefore, many terms may be needed to ensure convergence. A good initial estimate is 200 terms for each series.

The solution can yield results with either D,, D,, or h=O. Linear equilibrium adsorption and ion exchange can be simulated by dividing the coefficients D,, D,, D,, and V by the retardation factor, R (eq. 15). Temporal variations in solute concentration and odd- shaped source configurations can be simulated through the principle of superposition.

Description of program PATCHF

The program PATCHF computes the analytical solution to the three-dimensional solute-transport equation for an aquifer of finite width and height with a finite-width and finite-height solute source at the inflow boundary. It consists of a main program and subroutine CNRMLF. The functions of the main program and subroutine are outlined below; the program code listing is presented in attachment 2.

The program also calls the subroutine EXERFC and output subroutines TITLE, OFILE, PLOT3D, and CNTOUR, which are common to most programs described in this report. These subroutines are described in detail later.

Main program


The program next executes a set of four nested loops. The innermost loop calls subroutine CNRMLF to calculate the concentration at all specified y- coordinate values for a particular x-coordinate value, z-coordinate value, and time. The second loop cycles through all x-coordinate values. The third loop cycles through all z-coordinate values and prints a table of concentration in relation to x and y for each z value. The outer loop cycles through all specified time values. Model output can be plotted as a series of maps showing lines of equal solute concentration in the horizontal (x-y) plane at each point along the z-axis.

Subroutine CNRMLF

Subroutine CNRMLF calculates the normalized concentration (C/C,> for a particular time value and distance using equation 114. The maximum number of terms in the infinite series summation is specified by the user. Because terms in the series tend to oscillate, a subtotal of the last 10 terms is kept, and when the subtotal is less than a convergence criterion set at 1 x lo-“, the series summation is halted. If the series does not converge after the specified maximum number of terms are taken, a warning message is printed on the program output.


Table 9.-Input data format for the program PATCHF

Data Variable set Columns Format name Descrintion

1 1 - 60 A60 TITLE Data to be printed in a title box on the first page of program output. Last line in data set must have an "=- in column 1. First four lines are also used a5 title for plot.

------------------_--------------------------------------------------------------------~-------------------- 2 l- 4 14 Nx Number of x-coordinates at which solution will be evaluated.

S- 6 I4 NY Number of y-coordinates at which solution will be evaluated.

Q-12 14 NZ Number of s-coordinates at which solution will be evaluated.

13 - 16 14 NT Number of time values at which solution will be eva:Luated.

17 - 20 14 NMAX Maximum number of terms to be used in inner loop of the infinite series sumnation.

21 - 24 14 MAX Maximum number of terms to be used in outer loop of the infinite series sumnation.


------------------__--------------------------------------------------------------------~------------------- 3 1 - 10 A10 CUNITS Character variable used as label for units of concentration in program

output.




41 - 50 A10 LUNITS Unit5 of length.

51 - 60 A10 TUNITS Units of time. ------------------__--------------------------------------------------------------------~-------------------


11 - 20 F1O.O VX Ground-water velocity in x-direction.

21 - 30 F1O.O DX Longitudinal dispersion coefficient.


41 - 50 F10.0 DZ Transverse dispersion coefficient in s-direction.

51 - 60 F1O.O DK First-order solute-decay coefficient. ________________________________________-----------------------------------------------~--------------------

5 1 - 10 F1O.O W Aquifer width (aquifer extends from y - 0 to y = W).

11 - 20 F1O.O H Aquifer thickness (aquifer extends from s = 0 to s := El.

21 - 30 F1O.O Yl Y-coordinate of lower limit of patch solute source.

31 - 40 F10.0 Y2 Y-coordinate of upper limit of patch solute source.

41 - 50 F10.0 Zl Z-coordinate of lower limit of patch solute source.

51 - 60 F1O.O 22 Z-coordinate of upper limit of patch solute source. ____________________--------------------------------- _______________-_-----------------~--------------------

6 1 - 60 6FlO.O X(f) X-coordinates at which solution will be evaluated (eight values per line).

____________________-------------------------------------------------------~-------------------- 7 1 - 60 6FlO.O Y(I) Y-coordinates at which solution will be evaluated (eight values per

line). _________________-_------------------------------- ____________________------------------~-------------------

6 1 - 60 6FlO.O Z(I) Z-coordinates at which solution will be evaluated (eight values per line).

______-__-_--------------------------------- ____________________------------------------~------------------- Q 1 - 60 6FlO.O T(I) TimGn;lues at which solution will be evaluated (eight values per

. ____________________------------------------------------------------------ _________-----_--_---------------- 110 1 - 10 F10.0 XSCLP Scaling factor by which x-coordinate values are divided to convert them

to plotter inches.


21 - 30 F10.0 DELTA Contour increment for plot of normalized concentration (must be between 0.0 and 1.0).

'Data line is needed only if IPLT (in data set 21 is greater than 0.


Sample problem 10 $L,,$+D e+D,$-V$$hC (115) “ati

In sample problem 10, migration of chloride ion from a landfill, created by filling in a gravel pit excavated in a valley-fill aquifer, is simulated. Model variables are


c=c,, x=0 and Y,<y<Y, and Z,<z<Z, (116a)

Aquifer width (W) Aquifer height (H) Y-coordinate of lower limit of

source (Y J Y-coordinate of upper limit of

source (YJ Z-coordinate of lower limit of

source (Z,) Z-coordinate of upper limit of

source (Z,) Source concentration (C,) Ground-water velocity (V) Dispersion in x-direction (D,) Dispersion in y-direction (D,) Dispersion in z-direction (D,)

=3,000 ft =lOO ft

=400 ft

=2,000 ft

=50 ft

= 100 ft =l,OOO mg/L =l ft/d =200 ft’ld =60 ft’/d =lO ft’ld.

Concentrations are calculated at 150-ft intervals along the x-axis for 3,900 ft, and at lOO-ft intervals along the y-axis for 3,000 ft. Chloride concentration distributions after 3,000 days for z-coordinates of 50 and 75 ft (z=O is at the base of the aquifer) are simulated.

The input data set for sample problem 10 is shown in figure 20A; computer-generated contour plots of normalized concentration (C/C,) in x-y planes defined by the two z-coordinates are shown in figure 20B. The plot of concentrations along the centerline of the plume (at z=75 ft) can be compared with figure 13B to show the effect of vertical dispersion on both the shape of the chloride plume and simulated concentrations. This demonstrates the type of errors that can be introduced by using a two-dimensional solution when a three-dimensional solution is required.

Program output for sample problem 10 is presented in attachment 4. The sample problem required 7 min and 50 s of CPU time on a Prime model 9955 Mod II.

Aquifer of infinite width and hei ht with finite-width and finite-heig Ii t

solute source

Governing equation

Three-dimensional solute-transport equation:

c=o, x=0 and y<Y, or y>Y, and z<Z, or z>Z, (116b)

c aC,o ‘dy ’

y=*cc (117)

c a() ‘dz ’

z= kcc (118)

c aC,o ‘ax ’

x=co (119)


Y, =y-coordinate of lower limit of solute source, Y, =y-coordinate of upper limit of solute source, Z, =z-coordinate of lower limit of solute source, and Z, =z-coordinate of upper limit of solute source at

x=0.

Initial condition:

C=O, O<x<m, --oo<y<+m, and --oo<z<+m at t=O (120)

Assumptions :


formation (for a conservative solute, X=0>. 3. Flow is in x-direction only, and velocity is constant. 4. The coefficients of longitudinal dispersion (D,) and

transverse dispersion (D,, D,), from equation ‘7, are constant.

Analytical solution

Sagar (1982, p. 49) presents a solution to the anal- ogous problem of vertical leaching of a conservative solute from a patch source in the x-y plane. The following analytical solution was derived by the author using Fourier transforms (detailed derivation presented in attachment 1) for a patch source in the y-z plane with solute subject to decay:


I - WlM w!“bv -

L33.4 NI 'SIXtl-A 3NO-M 33NHlSIO

1333 NI ‘SIX&J. 3NOlt’ 33NklISIO 1333 NI ‘SIX&J. 3NOlt’ 33NklISIO


C(x,y,z,t)= C, x exp g [ 1 x WTD,

l {erfc[s]-erfc[$$]} l [erfc[$$]-erfc[$$]]dT. (121a)

where T is a dummy variable of integration for the time integral.

To improve the accuracy of the numerical integration, a variable substitution can be made where 7=Z4, yielding

m,Y,z,t)=

C, x exp g [ 1 x 2-\/43x

l j--$exp[ -(g++-&]

l [erfc[$$j=]-erfc[~]]dZ. (121b)

Comments:

The integral in equation 121b cannot be simplified further and must be evaluated numerically. A Gauss- Legendre numerical integration technique was used in the computer program written to evaluate the analytical solution and is described later. Round-off errors may still occur when evaluating the solution for very small values of x at late times.

Linear equilibrium adsorption and ion exchange can be simulated by dividing the coefficients D,, D,, D,, and V by the retardation factor, R (eq. 15). Temporal variations in solute concentration and odd-shaped source configurations can be simulated through the principle of superposition. A solution where the patch source is located in the x-y plane at z=O and velocity is in the x-direction can be found in Sagar (1982, p. 51).

Description of program PATCHI

The program PATCH1 computes the analytical solution to the three-dimensional solute-transport equation for an aquifer of infinite width and height with a finite-width and finite-height solute source at the inflow boundary. It consists of a main program and the subroutine CNRMLI. The functions of the main program and the subroutine are outlined below; the program code listing is presented in attachment 2.

The program also calls subroutines EXERFC and GLQPTS and the output subroutines TITLE, OFILE, and PLOT3D, which are common to most programs described in this report. These subroutines are described in detail later.

Main program


The program next executes a set of four nested loops. The innermost loop calls subroutine CNRMLI to calculate the concentration at all specified y- coordinate values for a particular x-coordinate value, z-coordinate value, and time. The second loop cycles through all x-coordinate values. The third loop cycles through all z-coordinate values and prints a table of concentration in relation to x and y for each z value. The outer loop cycles through all specified time values. Model output can also be plotted as a series of maps showing lines of equal solute concentration in the horizontal (x-y) plane at each point along the z-axis.

Subroutine CNRMLI

Subroutine CNRMLI calculates the normalized concentration (C/C,,) for a particular time value and distance. The integral in equation 121b is evaluated through a Gauss-Legendre numerical integration technique. The normalized roots of the Legendre polynomial and the corresponding weighting coefficients are passed by subroutine GLQPTS and scaled in the subroutine to account for the non-normalized limits of integration (from 0 to t1’4 rather than from - 1 to +1>.

The number of terms summed in the numerical integration (equivalent to the order of the polynomial) is specified by the user. Roots of the Legendre polynomial of order 4, 20, 60, 104, and 256 are provided in data file GLQ.PTS. In general, the more terms used in the integration, the more accurate the approximation; however, this must be weighed against the corresponding increase in computational effort and time.

TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS

Table IO.-input data format for the program PATCHI

Data ValZiflble set c01usms Format name Description

1 1 - 80 A80 TITLE Data to be printed in a title box on the first page of program output. Last line in data set must have an "a* in column 1. First four lines are also used as titles for plot.

-----------------------------------------------------------------------------------------~------------------ 2 l- 4 14 Nx Number of x-coordinates at which solution will be evaluated.


9-12 14 NZ Number of a-coordinates at which solution will be evaluated.


17 - 20 I4 NM4x Number of terms to be used in nmerical integration technique (must be equal to 4, 20, 80, 104, or 258).


-----------------------------------------------------------------------------------------~------------------ 3 1 - 10 A10 CUNITS Character variable used as label for units of concentration in program

output.

11 - 20 Al0 VUNITS Units of ground-water velocity.

21 - 30 Al0 DUXITS Units of dispersion coefficient.

31 - 40 Al0 KUNITS Units of solute-decay coefficient.

41- 50 A10 LUNITS Units of length.

51 - 80 A10 TUNITS Units of time. -----------------------------------------------------------------------------------------.-------------------


11 - 20 F1O.O vx Ground-water velocity in x-direction.

21 - 30 F1O.O DX Longitudinal dispersion coefficient.


41 - 50 F10.0 DZ Transverse dispersion coefficient in z-direction.

51 - 80 P10.0 DK First-order solute-decay coefficient. ________________________________________-------------------------------------------------~------------------ 5 1 - 10 F1O.O Yl Y-coordinate of lower limit of finite width and height. solute source.

11 - 20 P10.0 Y2 Y-coordinate of upper limit of finite width and height solute source.

21- 30 F1O.O Zl Z-coordinate of lower limit of finite width and height solute source.

31 - 40 F10.0 22 Z-coordinate of upper limit of finite width and height solute source. ________________________________________--------------------------------------------------.------------------ 8 1 - 80 8FlO.O X(I) X-coordinates at which solution will be evaluated (eight values per

line). _-__________________---------------------------------------------------------------------~--------------.---- 7 1 - 80 8FlO.O Y(I) Y-coordinates at which solution will be evaluated (ei&tt values per 1

line). ________________________________________-------------------------------------------------~--------------~--- 8 1 - 80 8FlO.O Z(I) Z-coordinates at which solution will be evaluated (eight values per

line). ________________________________________-------------------------------------------------,------------------- 9 1 - 80 9PlO.O T(I) Time values at which solution will be evaluated (eight values per

line). ________________________________________-------------------------------------------------~------------- ----- 110 1 - 10 F10.0 XSCLP Scaling factor by which to divide x-coordinate values are divided to

convert them to plotter inches.


21 - 30 F1O.O DELTA Contour increment for plot of normalized concentration Gnust be between 0.0 and 1.0).



Additional discussions of the numerical integration technique are presented in a later section describing subroutine GLQPTS.

Sample problem 11

In sample problem 11, a contaminant plume containing “Sr (strontium-90) from a deep radioactive-waste storage facility migrates through a thick, confined aquifer. Model variables are

Source width (W,) =1,200 ft Source height (H,) = 300 ft Y-coordinate of lower limit of source (YJ = 900 ft Y-coordinate of upper limit of source (YJ =2,100 ft Z-coordinate of lower limit of source (Z,) =1,350 ft Z-coordinate of upper limit of source (Z,) =1,650 ft Ground-water velocity (V) =l ft/d Longitudinal dispersivity (a,) = 100 ft Transverse dispersivity (at) = 2oft Source concentration (C,) =lOO mg/L Half-life of “Sr =28 years.


Coefficient of longitudinal dispersion (D,) =lOO ft2/d

Coefficients of transverse dispersion (DY and D,) = 20 ft21d

First-order solute-decay coefficient (A) =6.78x lop5 per day.

Concentrations are calculated at 150-ft intervals along the x-axis for 3,900 ft, and at lOO-ft intervals along the y-axis for 2,600 ft. The “Sr concentration distribution after 10 years (3,652.5 days) for z- coordinates of 1,650, 1,700, and 1,750 ft (z=O at the base of the aquifer and z=1,650 at the top of the storage facility) is simulated.

The input data set for sample problem 11 is shown in figure 21A; computer-generated contour plots of the normalized concentration (C/C,,) in x-y planes defined by the three z-coordinates are shown in figure 21B. Program output for this sample problem is presented in attachment 4. Sample problem 11 required 3 min 20 s of CPU time on a Prime model 9955 Mod II.

Description of Subroutines

The subroutines described in this section are common to most of the programs developed to evaluate the analytical solutions. Subroutines EXERFC and GLQPTS are used in evaluating terms in the analytical solutions, OFILE and TITLE are used in program

input and output, and PLOTlD, PLOTBD, PLOT3D, and CNTOUR are used to graphically display program results. Subroutine listings are presented in attachment 3.

Mathematical subroutines

Subroutines EXERFC and GLQPTS

Subroutine EXERFC is called to evaluate the product of an exponential and complementary error function (exp[x]* erfc[y]), where the error function, erf(y), is defined as

(122)

and the complementary error function, erfc(y), is defined as

erfc(y)=l.O-erf(y). (123)

Often, the values of x and y are such that erfc(y) is very small (less than lx lo-l2 for y=5), whereas exp(x) is very large. To accurately calculate the product of the two functions, a high degree of accuracy is needed in the calculation of erfc(y). Subroutine EXERFC uses a rational Chebyshev approximation (Cody, 1969), accurate to between 10 and 13 significant figures, to calculate erf(y) or erfc(y). The two variables x and y are passed to the subroutine. To calculate only erfc(y), the routine EXERFC can be called with the value of x set to zero.

For absolute values of y less than 0.469, the rational Chebyshev approximation is given by

erf(y)= yr P&p/r Qlip, (124) i=O i=O

where Pl and Ql are the coefficients of the rational approximation given by Cody (1969) for n=5. For negative values of y, the symmetry condition that erf(-y)= -erf(y) (Abramowitz and Stegun, 1964) is used. Erfc(y) is then given by equation 123.

For absolute values of y between 0.469 and 4.0, a rational approximation for erfc(y) is used, given by

(125) i=O i=O

where P2 and Q2 are the coefficients given by Cody (1969) for n=8. For negative values of y, the identity that erfc(-y)=2-erfc(y) is used.

For absolute values of y greater than 4.0, a second rational approximation for erfc(y) is used, given by


where P3 and Q3 are coefficients given by Cody (1969) for n=5. When a product of exp(x) and erfc(y) is calculated, the arguments for the exponential in equations 125 and 126 are changed to (x-f).

Subroutine GLQPTS is called to numerically evaluate the time integral found in several of the analytical solutions. The Gauss integration formula used is given by Abramowitz and Stegun (1964) as

n

(127)

where zi =roots of Legendre polynomial for a particular

value of n, and wi =corresponding weighting functions. Positive roots of the Legendre polynomials for n=4,

20, 60, 104, and 256 and their weighting functions, as given in Cleary and Ungs (1978), have been tabulated and are read from a data file called GLQ.PTS. Sub- routine GLQPTS calculates the negative roots and their weighting coefficients. These values are passed to the other subroutines through an array in common. A listing of file GLQ.PTS is presented in attachment 3.

As stated earlier, the accuracy of the numerical integration is increased if the user selects a larger value for n. However, computational effort is also increased. Checks can be made to determine whether a smaller value for n produces reasonable results by comparing the solution for a particular n with that obtained using the next higher value. Roots and weighting coefficients for additional values of n can be found in Abramowitz and Stegun (1964, p. 916-919).

.The subroutine is set up to read data file GLQ.PTS on logical unit 77 on the Prime system. For systems other than Prime, this routine should be modified to include the correct system-dependent file opening statements. Also, file-naming conventions for the particular system must be observed, and the data file renamed appropriately.

Input/Output subroutines

Subroutines OFILE and TITLE

Subroutine OFILE is used to open disk files for program input and output on the Prime computer system. It assigns logical unit 15 to the input data file and logical unit 16 to the file for program output. The user is queried at the terminal (logical unit 1) for the name of the appropriate disk files, and any file name up to 50 characters in length can be entered. For

output to be sent directly to the terminal, the user should type an asterisk (*) in column 1 when asked for the output file name.

For systems other than the Prime, this routine should be modified to include the correct system- dependent file opening statements. Also, the logical units (1, 15, and 16) should be changed if they are not appropriate for the particular system.

Subroutine TITLE is called by all programs to print a title box on the first page of model output. Titles are supplied as the first lines of the input data set. Titles are automatically centered, and the routine closes the title box when it encounters an equal sign (=> in column 1 of a data line. The routine also prints the date and time the program execution began. The first four title lines are used as titles for plots.

Subroutine TITLE calls the Prime-supplied functions TIME$A and DATE$A found in the library VAPPLB. For non-Prime systems, these calls should be modified or, if similar functions are not available, deleted.

Graphics subroutines Four subroutines, PLOTlD, PLOT2D, PLOT3D,

and CNTOUR, were developed to graphically display selected output from the programs described in this report. These subroutines contain calls for DISSPLA graphics software (Integrated Software Systems Cor- poration, 1981>, and the DISSPLA library must be loaded when compiling the programs. Users who do not have access to DISSPLA software can easily modify the DISSPLA software calls to those appropriate to their own graphics software.

Subroutines PLOTlD, PLOTBD, and PLOT3D contain a call to COMPRS, which creates a META file that can be output, at a later time, to a wide variety of plotter devices through the DISSPLA postprocessor. This call can be replaced with a call to directly nominate a plotter device (such as a graphics terminal) so that plots can be drawn as the programs execute. The user should consult the DISSPLA users manual (Integrated Software Systems Corporation, 1981) for more information.

Subroutines PLOT1 D, PLOTZD, PLOT3D, and CNTOUR

Subroutine PLOTlD is called by the programs FINITE and SEMINF to create plots of the normalized concentration C/C, in relation to distance for each of the time values specified in the input data. An example of typical plotter output is shown in figure 4B. DISSPLA software calls are used to draw the axes and to plot the data points. The height of the plot is 12.5 in. The width is controlled by the difference


between the minimum and maximum x-coordinate value and by the scale factor XSCLP specified in data set 4 (tables 1, 2). If no plotter is available, the user can either specify a value of 0 for IPLT in data set 2 (tables 1, 2) or delete the call to PLOTlD in the main programs of FINITE and SEMINF.

Subroutine PLOTZD is called by the programs POINTB, STRIPF, STRIPI, and GAUSS to initialize a plot of lines of equal normalized concentration (C/C,) in the x-y plane for each of the specified time values. A typical example is shown in figure 13B.

The size of each subplot depends on the difference between the maximum and minimum x- and y- coordinate values and the plot scaling factors XSCLP and YSCLP specified by the user in data set 9 (tables 3-5). The overall length of the plot is determined by the number of time values specified. The contour increment DELTA (a value between 0.0 and 1.0) is specified by the user in data set 9.

Subroutine PLOTBD defines the plot and subplot sizes, draws and labels the axes, and then calls subroutine CNTOUR, which draws and labels the contours. If no plotter is available, IPLT in data set 2 (tables 3-6) can be set to 0, or the call to PLOTBD in the main programs STRIPF, STRIPI, and GAUSS can be deleted.

Subroutine PLOTSD is called by the programs POINT3, PATCHF, and PATCH1 to initialize a plot of lines of equal normalized concentration (C/C,,) in the x-y plane for each of the z-coordinates and time values specified in the input data. An example of plotter output from this subroutine is shown in figure 20B.

The size of each subplot depends on the difference between the maximum and minimum x- and y- coordinates and the plot scaling factors XSCLP and YSCLP specified by the user in data set 10 (tables 8-10). The overall length of the plot is determined by the number of z-values specified. Separate plots are drawn for each specified time value. The contour increment DELTA (a value between 0.0 and 1.0) can also be specified by the user, in data set 10.

Subroutine PLOT3D defines the plot and subplot sizes, draws and labels the axes, and then calls subroutine CNTOUR, which draws and labels the contours. If no plotter is available, IPLT in data set 2 (tables S-10) can be set to zero, or the call to PLOT3D in the main programs of POINT3, PATCHF, and PATCH1 can be deleted.

Subroutine CNTOUR is called to produce simplified plots of lines of equal normalized concentration (C/C,,) in the x-y plane for each of the time values or z-coordinates specified in the input data. Although there are many software packages that contour grid- ded data, such as concentration in relation to x and y, some of these require the grid to be equally spaced

and others, such as that contained in DISSPLA, can interpolate scattered data onto regular grids, but at the cost of considerable computational effort and time.

The subroutine first creates a rectangular grid based on the x- and y-coordinates supplied in the input data. Each rectangular block in the grid is then subdivided into two triangles defined by a diagonal drawn across the block. Next, contour segments are drawn by connecting points of equal concentration determined by linear interpolation along the axes of each triangular element.

The number of contours drawn is determined by the difference between the maximum and minimum normalized concentration values and the contour increment, DELTA. The subroutine uses a relatively complex algorithm to connect the contour segments defining a contour line and to determine whether a contour line has exited the grid or formed a closed loop. Contour lines are labeled after all NUM contour segments are drawn. NUM is set. to 40 in the code, but this can be changed by the user. The routine requires three work arrays--PC, YPC, and IFL,4G-to store contouring data. IFLAG must be dimensioned to twice the number of rectangular blocks. XPC and YPC are dimensioned by 50 in the subroutine and in common block PDAT in the main programs. This number must be changed if the user increases the value of NUM to greater than 50.

Running the programs Array dimensions

Dimensions of arrays used by the programs are set by a PARAMETER statement, as follows:

PARAMETER (MAXX=lOO, MAXY=50, MAXZ =30, MAXT=20, MAXXY=5000, MAXXYZ=:lOOOO,

MAXRT= 1000) where

MAXX =maximum number of x-coordinates, MAXY =maximum number of y-coordinates, MAXZ =maximum number of z-coordinates, MAXT =maximum number of time values,

MAXXY =product of MAXX ;md MAXY, MAXXYB = twice MAXXY, and

MAXRT =maximum number of roots used in series summation in program FINITE.

The user can modify the PARAMETER statement to increase or decrease these limits.

Compiling and loading

The following describes the procedure for compiling and running the programson the Prime system. For convenience, the user should first create a single file


called SUBS.F77 that contains the following subroutines: EXERFC, GLQPTS, OFILE, TITLE, PLOTlD, PLOTZD, PLOT3D, AND CNTOUR. The user should then type

Fi’7 PROGRAM.F77 -BIG -SILENT F77 SUBSF77 -BIG -SILENT SEG -LOAD LOAD PROGRAM LOAD SUBS LIBRARY DISSPLA LIBRARY VAPPLB LI SAVE QUIT

where PROGRAM indicates the name of the main program (for example, STRIPF or GAUSS). After the message “LOAD COMPLETE” is received at the terminal, the user can run the program by typing

SEG PROGRAM

The following message will appear

“TYPE IN INPUT FILE NAME”

The user can respond with the name of the file containing the data set (see description of subroutine OFILE). The following will then appear:

“TYPE IN OUTPUT FILE NAME”

The user can respond with the name of the output file name or an asterisk (*> to cause output to come to the terminal.

These programs can be run on other computer systems, although some device-dependent subroutine calls may have to be modified. These statements are identified in the previous section.

Summary

The physical, chemical, and biological processes that govern transport of solutes in ground water can be described quantitatively by the advective- dispersive solute-transport equation. Analytical solutions, which are exact mathematical solutions for this partial differential equation, have been derived for many combinations of aquifer geometry, solute-source configurations, and boundary and initial conditions. These solutions can be used to mathematically model the movement of solutes in homogeneous aquifers having simple flow systems in which the chemical and biological processes can be described by linear relations.

This report presents analytical solutions for solute transport in one-, two-, and three-dimensional systems having uniform flow. The solutions were compiled from those published in various journals and reports or were derived by the author. The solutions for one-dimensional solute transport are for (1) a finite-length system with a first-type boundary condition at the inflow end, (2) a finite-length system with a third-type boundary condition at the inflow end, (3) a semi-infinite system with a first-type boundary condition at the inflow end, and (4) a semi-infinite system with a third-type boundary condition at the inflow end. Solutions for the finite-length system assume a second-type boundary condition at the outflow end.

Solutions for two-dimensional solute transport were presented for (1) an aquifer of infinite area1 extent with a continuous point source at which fluid is injected at a constant rate and concentration, (2) a semi-infinite aquifer of finite width with a strip source along the inflow boundary, (3) a semi-infinite aquifer of infinite width with a strip source along the inflow boundary, and (4) a semi-infinite aquifer of infinite width with a solute source having a gaussian concentration distribution. Solutions for three-dimensional solute transport were presented for (1) an aquifer of infinite extent with a continuous point source, (2) a semi-infinite aquifer of finite height and width with a patch source along the inflow boundary, and (3) a semi-infinite aquifer of infinite width and height with a patch source along the inflow boundary. All the solutions presented can account for first-order solute decay due to chemical or biological processes and linear equilibrium adsorption.

A set of computer programs was written to evaluate these solutions and to produce tables and graphs of solute concentration as a function of time and distance from the solute source. Documentation of these programs includes instruction on their use, description of input data format, sample problems, and sample data sets. Source codes for the programs and output for the sample problems are presented in attachments to the report.

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